ANALYSIS AND PDE ON METRIC MEASURE
SPACES: SOBOLEV FUNCTIONS AND VISCOSITY
SOLUTIONS
by
Xiaodan Zhou
B.S. in Mathematics, Beijing Normal University, 2011
Submitted to the Graduate Faculty of
the Dietrich School of Arts and Sciences in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
University of Pittsburgh
2016
UNIVERSITY OF PITTSBURGH
DIETRICH SCHOOL OF ARTS AND SCIENCES
This dissertation was presented
by
Xiaodan Zhou
It was defended on
June 16th, 2016
and approved by
Piotr Haj lasz, Department of Mathematics, University of Pittsburgh
Juan J. Manfredi, Department of Mathematics, University of Pittsburgh
Giovanni P. Galdi, Department of Mechanical Engineering and Materials Science,
University of Pittsburgh
Christopher J. Lennard, Department of Mathematics, University of Pittsburgh
Dissertation Advisors: Piotr Haj lasz, Department of Mathematics, University of
Pittsburgh,
Juan J. Manfredi, Department of Mathematics, University of Pittsburgh
ii
Copyright c© by Xiaodan Zhou
2016
iii
ABSTRACT
ANALYSIS AND PDE ON METRIC MEASURE SPACES: SOBOLEV
FUNCTIONS AND VISCOSITY SOLUTIONS
Xiaodan Zhou, PhD
University of Pittsburgh, 2016
We study analysis and partial differential equations on metric measure spaces by investigating
the properties of Sobolev functions or Sobolev mappings and studying the viscosity solutions
to some partial differential equations.
This manuscript consists of two parts. The first part is focused on the theory of Sobolev
spaces on metric measure spaces. We investigate the continuity of Sobolev functions in the
critical case in some general metric spaces including compact connected one-dimensional
spaces and fractals. We also construct a large class of pathological n-dimensional spheres
in Rn+1 by showing that for any Cantor set C ⊂ Rn+1 there is a topological embedding
f : Sn → Rn+1 of the Sobolev class W 1,n whose image contains the Cantor set C.
The second part is focused on the theory of viscosity solutions for nonlinear partial dif-
ferential equations in metric spaces, including the Heisenberg group as an important special
case. We study Hamilton-Jacobi equations on the Heisenberg group H and show uniqueness
of viscosity solutions with exponential growth at infinity. Lipschitz and horizontal convexity
preserving properties of these equations under appropriate assumptions are also investigated.
In this part, we also study a recent game-theoretic approach to the viscosity solutions of var-
ious equations, including deterministic and stochastic games. Based on this interpretation,
we give new proofs of convexity preserving properties of the mean curvature flow equations
and normalized p-Laplace equations in the Euclidean space.
iv
Keywords: Sobolev spaces, metric measure spaces, Heisenberg group, viscosity solution,
convexity preserving, game-theoretic methods.
v
TABLE OF CONTENTS
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1.0 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2.0 REVIEW OF METRIC MEASURE SPACES . . . . . . . . . . . . . . . . 10
2.1 Basic definitions in metric measure spaces . . . . . . . . . . . . . . . . . . . 10
2.2 Curves in metric measure spaces . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Rectifiability of one-dimensional compact and connected spaces . . . . . . . 16
3.0 SOBOLEV FUNCTIONS ON METRIC MEASURE SPACES . . . . . 19
3.1 Sobolev functions in the Euclidean space . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . 19
3.1.2 Equivalent characterizations . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Sobolev functions on the metric measure spaces . . . . . . . . . . . . . . . . 24
3.2.1 Sobolev spaces M1,p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Sobolev spaces P 1,p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.3 Sobolev spaces N1,p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Spaces supporting the Poincare inequality . . . . . . . . . . . . . . . . . . . 29
4.0 SOBOLEV FUNCTIONS IN THE CRITICAL CASE ON METRIC
MEASURE SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1 On Compact, connected metric spaces with H1(X) <∞ . . . . . . . . . . . 33
4.1.1 Absolutely continuous functions on the metric spaces . . . . . . . . . 33
4.1.2 Absolutely continuous characterization of Sobolev functions . . . . . . 34
4.1.3 Spaces supporting the Poincare inequality . . . . . . . . . . . . . . . . 39
4.2 On s-Ahlfors regular metric spaces with s ≤ 1 . . . . . . . . . . . . . . . . . 41
vi
5.0 SOBOLEV EMBEDDING OF A SPHERE CONTAINING AN ARBI-
TRARY CANTOR SET IN THE IMAGE . . . . . . . . . . . . . . . . . . 46
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 Cantor sets and trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2.1 Ternary Cantor set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2.2 Cantor trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 Sobolev tentacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.4 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.5 Uncountable inequivalent embeddings . . . . . . . . . . . . . . . . . . . . . 62
6.0 SEMILINEAR EVOLUTION EQUATIONS IN THE HEISENBERG
GROUP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.1 The Heisenberg group H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.1.1 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . 66
6.1.2 Metrics on H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.1.3 Lipschitz continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.1.4 Horizontal convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 Review of viscosity solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.2.1 Viscosity solutions in Rn . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.2.2 Viscosity solutions in H . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3 Uniqueness of unbounded solutions . . . . . . . . . . . . . . . . . . . . . . . 78
6.4 Lipschitz preserving properties . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.4.1 Right invariant Lipschitz continuity preserving . . . . . . . . . . . . . 87
6.4.2 A special class of Hamilton-Jacobi equations . . . . . . . . . . . . . . 89
6.5 Convexity preserving properties . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.5.1 Right invariant h-convexity preserving . . . . . . . . . . . . . . . . . . 93
6.5.2 Left invariant h-convexity preserving . . . . . . . . . . . . . . . . . . . 100
6.6 More examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.0 A GAME-THEORETIC PROOF OF CONVEXITY PRESERVATION
OF SOME NONLINEAR DIFFERENTIAL EQUATIONS . . . . . . . . 103
7.1 Convexity preserving properties . . . . . . . . . . . . . . . . . . . . . . . . . 103
vii
7.2 Game interpretations of viscosity solutions . . . . . . . . . . . . . . . . . . . 105
7.2.1 Tug of War game and normalized parabolic p-Laplace equations . . . 105
7.2.2 The deterministic game and the mean curvature flow equations . . . . 107
7.2.3 A modified game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.3 Game-theoretic proofs of convexity-preserving properties . . . . . . . . . . . 116
7.3.1 Convexity-preserving for the solutions to the normalized parabolic p-
Laplace equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.3.2 Convexity-preserving for level sets of the mean curvature flow equations 118
7.3.3 Convexity-preserving for the solutions to the mean curvature flow equa-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3.4 Convexity-preserving with Neumann boundary condition of the mean
curvature flow equations . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.3.4.1 Billiard games for the Neumann condition . . . . . . . . . . . 125
7.3.4.2 Convexity of level sets . . . . . . . . . . . . . . . . . . . . . . 126
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
viii
PREFACE
I would like to express my deepest gratitude to my advisor Piotr Haj lasz. I benefited a
lot both from the enlightening discussions with him on research and from the wonderful
classes he taught. He provided a lot of useful suggestions and comments on my research
projects and great support to explore new topics. His expertise, understanding, guidance
and encouragement made it possible for me to complete my PhD study.
I would also like to extend my greatest gratitude to my advisor Juan Manfredi for his
guidance and helpful advices in both mathematics and career. His insight and enthusiasm
are very inspiring for me. I am also very thankful to his kindness and support. It is a great
pleasure to work with him.
I am hugely indebted to Qing Liu, my friend, partner and collaborator, for introducing
a new field of research to me, for his tremendous help, for his unwavering support and being
a source of motivations.
I am also especially grateful to Nageswari Shanmugalingam for her constant support and
many stimulating discussions.
I would like to thank all my committee members and my fellow graduate students who
gave many informative talks during our seminars.
Last but not least, I would like to thank my family for their love and constant faith on
me and my friends for their company and the joys they brought to me.
ix
1.0 INTRODUCTION
Analysis on metric measure spaces is a new area in contemporary mathematics that has been
developed since the nineties. Because of its general setting it plays a fundamental role in the
unification of methods that previously have been developed separately for different areas of
mathematics [4, 5, 55, 59]. Besides the pure mathematical importance of these problems,
analysis on metric spaces has been widely applied to image reconstruction theory, optimal
transport, control theory, robotics and mathematical biology.
In the first part of this manuscript, we study the Sobolev functions on general metric
measure spaces. The theory of Sobolev spaces is one of the main tools in analysis on metric
spaces. Let (X, d, µ) denote a metric space equipped with a doubling measure µ. Given a
Borel function u : X → R, we say that a Borel function g : X → [0,∞] is an upper gradient
of u if
|u(γ(a))− u(γ(b))| ≤ˆγ
g
for every rectifiable curve γ : [a, b] → X. The Newtonian-Sobolev space N1,p(X) is defined
as the collection of all p-integrable functions with p-integrable upper gradients [112]. The
notion of upper gradient is an important generalization of the length of the gradient.
While the N1,p spaces can be defined on general metric measure spaces, without ad-
ditional information about the structure of the underlying metric space the theory is not
interesting. Indeed, if there are no rectifiable curves in the space then N1,p = Lp. On the
other hand the theory of N1,p spaces is very rich when X supports the so called p-Poincare
inequality. Spaces supporting Poincare inequalities introduced by Heinonen and Koskela
[56], provide a good structure to study first-order analysis. Metric measure spaces support-
ing an abstract Poincare inequality are highly connected. Spaces like fractals with limited
1
connectedness do not belong to this class. However, in this case, one can define the Haj lasz-
Sobolev space M1,p(X) as the collection of all p-integrable functions for which there exist
nonnegative g ∈ Lp(X) such that
|u(x)− u(y)| ≤ d(x, y)(g(x) + g(y)
), (1.0.1)
for x, y ∈ X \ E where µ(E) = 0. We denote the collection of nonnegative Borel function g
as above by D(u).
There are many other extensions of the classical theory of Sobolev spaces to the settings
of metric measure spaces [4, 5, 24, 46, 49, 112]. Under suitable assumptions, some of or all
these Sobolev spaces defined via different approaches coincide [59, 74]. We will review some
basic definitions and theorems of Sobolev functions on metric measure spaces in Chapter 2
and Chapter 3.
In the Euclidean space, Sobolev functions u ∈ W 1,1([a, b]) are absolutely continuous.
When n ≥ 2, u(x) = log | log |x|| provides an example of a discontinuous Sobolev functions
in W 1,n(Bn(0, e−1)) in the n-dimensional space. We are interested in investigating the con-
tinuity of the Sobolev functions in the critical case, i.e., p = n, on metric measure spaces.
From the above examples, we conjecture that the Sobolev functions in the critical case may
lose the continuity if the Hausdorff dimension is greater than 1. In Chapter 4, we first gen-
eralize the characterization of the Sobolev functions by absolute continuity to some metric
spaces X that are compact, connected and have finite one-dimensional Hausdorff measure.
Note that there are many extensions of the definitions of absolutely continuous functions to
more general settings and much research has been done on the connections between these
absolutely continuous functions and Sobolev functions. Maly [93] extended this definition
to n-dimensional Euclidean space by replacing the pairwise disjoint subintervals [ai, bi] with
pairwise disjoint balls Bi and |u(bi) − u(ai)| with (osc(u(Bi)))n. Kinnunen and Tuominen
[76, Theorem 4] proved that on a doubling space X functions in the Haj lasz Sobolev space
M1,1(X) [46] coincide with Holder continuous functions outside a set of small Hausdorff
content.
The definition of classical absolutely continuous functions can be modified in the following
way in our settings. Recall that a simple curve is a curve without self intersections. In what
2
follows Hs stands for the s-dimensional Hausdorff measure.
Definition 1.0.1 ([121]). Let (X, d,H1) be a compact, connected metric measure space
with H1(X) < ∞. A function u : X → R is said to be absolutely continuous (denoted
u ∈ AC(X)) if for any ε > 0, there is a positive number δ such that
∑i
|u γi(`i)− u γi(0)| < ε,
for any finite collection of pairwise disjoint arc-length parametrized simple curves γi : [0, `i]→
X satisfying∑
i `i < δ.
In the real line, absolutely continuous functions are differentiable almost everywhere and
the fundamental theorem of calculus holds. For a compact, connected metric space X with
H1(X) <∞, there exists a good parametrization (Theorem 2.3.3) that can decompose this
space as a countable union of pairwise disjoint simple curves and a set with 1-dimensional
Hausdorff measure zero [5, Theorem 4.4.8]. We have the following result.
Theorem 1.0.2 ([121]). Let (X, d,H1) be a compact, connected metric measure space with
H1(X) <∞. If u ∈ AC(X), then there is an upper gradient g ∈ L1(X) of u, such that, for
any rectifiable curve γ : [a, b]→ X, we have
|u(γ(a))− u(γ(b))| ≤ˆγ
g.
For an absolutely continuous function u ∈ AC(X), if the upper gradient g in Theorem
1.0.2 is p-integrable, we write u ∈ ACp(X). By definition, it follows that ACp(X)∩Lp(X) ⊂
N1,p(X). On the other hand, we can verify that all Sobolev functions in N1,p(X) belong to
the class AC(X) and the associated upper gradients g are p-integrable. Thus, we have the
following result.
Theorem 1.0.3. Let (X, d,H1) be a compact and connected metric measure space with
H1(X) <∞. Then u ∈ N1,p(X) if and only if u ∈ ACp(X) and u ∈ Lp(X). In other words,
N1,p(X) = ACp(X) ∩ Lp(X).
3
We also prove that if a compact metric space is quasiconvex and 1-Ahlfors regular, then
X supports p-Poincare inequality for 1 ≤ p <∞. Recall that a metric space X is s-Ahlfors
regular if there exists a constant C0 ≥ 1 such that C−10 rs ≤ µ(B(r)) ≤ C0r
s for any ball
B(r) ⊂ X with 0 < r < diam(X). A metric space is quasiconvex if there exists a constant C
such that any two points x, y ∈ X can be joined by a rectifiable curve of length bounded by
Cd(x, y). It follows that absolutely continuous functions with p-integrable upper gradient
can be identified with the Sobolev functions defined via different approaches.
Theorem 1.0.4 ([121]). Let (X, d,H1) be a compact, quasiconvex and 1-Ahlfors regular
metric measure space. Let 1 < p <∞. Then
ACp(X) ∩ Lp(X) = N1,p(X) = P 1,p(X) = M1,p(X).
This result extends the characterization of Sobolev functions by absolute continuity
known in the 1-dimensional intervals to more general 1-dimensional spaces. In the next
theorem, we deal with some fractal type metric spaces. We prove the uniform continuity of
Sobolev functions in s-Ahlfors regular spaces with s ≤ 1. Note that Ahlfors regular spaces
include a large class of fractals generated by iterated function systems satisfying the open
set condition.
Theorem 1.0.5 ([122]). Let (X, d,Hs) be an s-Ahlfors regular metric space and 0 < s ≤ 1.
If u ∈M1,s(X, d,Hs), then u is uniformly continuous. Moreover, there exists a constant C,
such that for any ball B ⊂ X,
oscB |u| = supx,y∈B
|u(x)− u(y)| ≤ C(ˆ
2B
gsdHs) 1
s,
where g ∈ D(u) ∩ Ls(X).
In Chapter 5, we generalized a construction of the famous Alexander horned sphere,
which provides a counterexample to the Schoenflies Theorem in R3. Namely we managed
to construct a Sobolev embedding from the n-dimensional sphere to the (n+ 1)-dimensional
Euclidean space whose image contains an arbitrary Cantor set. Recall that a Cantor set
is any compact, totally disconnected and perfect set. The main theorem can be stated as
follows.
4
Theorem 1.0.6 ([52]). For any Cantor set C ⊂ Rn+1, n ≥ 2, there is an embedding
f : Sn → Rn+1 such that
(a) f ∈ W 1,n(Sn,Rn+1),
(b) C ⊂ f(Sn),
(c) f−1(C) ⊂ Sn is a Cantor set of Hausdorff dimension zero,
(d) f is a smooth diffeomorphism in Sn \ f−1(C).
Note that such a sphere f(Sn) can be very pathological due to topological complexity
of wild Cantor sets in Rn+1. Note also that the Cantor set C may have positive (n + 1)-
dimensional Lebegue measure. The Sobolev regularity of such a homeomorphism has not
been known previously. Moreover our construction generalizes not only the Alexander horned
sphere, but also provides a new class of Sobolev homomorphisms without the Luzin property.
We also prove in Theorem 5.5.1 that there are uncountably many such embeddings f : S2 →
R3 of class W 1,2(S2,R3) which are not equivalent.
In the second part of this manuscript, we study viscosity solutions to some partial differ-
ential equations. Since 1980s, the theory of viscosity solution for nonlinear partial differential
equations has been developed and applied to a wide range of fields. It provides existence
and uniqueness of weak solutions to a very general class of fully nonlinear equations in the
space of continuous functions [27, 80]. A recent trend is to extend this theory to metric mea-
sure spaces; see [3, 40] for some results of first order Hamilton-Jacobi equations on general
metric spaces. We study viscosity solutions to a class of second order equations. We focus
on the setting of sub-Riemannian manifolds [21], which not only play a very important role
in general analysis on metric spaces but also has applications in many other fields including
robotic control, neuroscience and digital image reconstruction. We aim to develop the vis-
cosity solution theory for nonlinear parabolic equations in the Heisenberg group H, which is
known as the simplest example of sub-Riemmanian manifold.
In Chapter 6, we review some basic definitions and properties of Heisenberg group and
the theory of viscosity solutions. Then we study viscosity solutions of the following semilinear
parabolic equationsut − tr(A(∇2
Hu)∗) + f(p,∇Hu) = 0 in H× (0,∞), (1.0.2)
u(·, 0) = u0 in H, (1.0.3)
5
in the Heisenberg group, where A is a given 2 × 2 symmetric positive-semidefinite matrix
and the function f : H× R2 → R satisfies the following Lipschitz assumptions.
(A1) There exists L1 > 0 such that
|f(p, w1)− f(p, w2)| ≤ L1|w1 − w2| (1.0.4)
for all p ∈ H and w1, w2 ∈ R2.
(A2) There exists L2(ρ) > 0 depending on ρ > 0 such that
|f(p, w)− f(q, w)| ≤ L2(ρ)|p · q−1|G (1.0.5)
for all p, q ∈ H with |p|, |q| ≤ ρ and all w ∈ R2.
Here | · |G denotes the Koranyi gauge in H,
|p|G =((x2
p + y2p)
2 + 16z2p
) 14
for all p = (xp, yp, zp) ∈ H. We first show uniqueness of viscosity solutions to the above
equation with exponential growth at infinity.
Theorem 1.0.7 (Uniqueness of solutions, [89]). Assume that (A1) and (A2) hold. Let
u0 ∈ C(H). Then there is at most one continuous viscosity solution u of (1.0.2)–(1.0.3)
satisfying the following exponential growth condition at infinity:
(G) For any T > 0, there exists k > 0 and CT > 0 such that |u(p, t)| ≤ CT ek〈p〉 for all
(p, t) ∈ H× [0, T ].
Among many properties of the viscosity solutions, the Lipschitz and convexity preserving
properties are known to be important for various linear and nonlinear parabolic equations
arising in geometry, material sciences and image processing. Suppose that u : Rn× [0,∞)→
R is a solution of a certain parabolic equation with initial condition u0. The property reads
as follows: when u0 is Lipschitz continuous (resp. convex), the unique solution u(x, t) is
Lipschitz (resp., convex) in x as well for any t ≥ 0. The convexity preserving property of
viscosity solutions was proved to hold in a very general class of degenerate parabolic equations
[39]. It is natural to ask whether the Lipschitz and convexity preserving properties also hold
6
for nonlinear parabolic equations in the Heisenberg group. Note that the notion of convexity
of functions in the Heisenberg group is known [30, 91]. More precisely, a function u is said
to be Lipschitz continuous in H if there exists L > 0 such that
|u(p)− u(q)| ≤ LdL(p, q)
for all p, q ∈ H, and horizontally convex in H if
u(p · h−1) + u(p · h) ≥ 2u(p)
for any p ∈ H and any h ∈ H0, where
H0 = h ∈ H : h = (h1, h2, 0) for h1, h2 ∈ R.
It turns out that in general such properties cannot be expected in the Heisenberg group.
Some restrictions on the class of solutions proved to be necessary. In fact, we obtained the
Lipschitz continuity and convexity preserving properties with respect to the right invariant
metric dR(p, q) = |p · q−1|G, which is invariant only under right translations and therefore
not equivalent to the usual gauge metric give by dL(p, q) = |p−1 · q|G.
Let us also present our results in a simple case. A more general version of Lipschitz
preserving is given in Theorem 6.4.2.
Theorem 1.0.8 (Preservation of right invariant Lipschitz continuity, [89]). Assume that
f : R2 → R is Lipschitz. Let u ∈ C(H× [0,∞)) be the unique solution of
ut − tr(A(∇2Hu)∗) + f(∇Hu) = 0 in H× (0,∞), (1.0.6)
with u(·, 0) = u0(·) satisfying the growth condition (G). If there exists L > 0 such that
|u0(p)− u0(q)| ≤ LdR(p, q)
for all p, q ∈ H, then
|u(p, t)− u(q, t)| ≤ LdR(p, q)
for all p, q ∈ H and t ≥ 0.
7
For the case of first order Hamilton-Jacobi equations (A = 0), if in addition we assume
that f : R2 → R is in the form that f(ξ) = m(|ξ|) with m : R → R locally uniformly
continuous, then the Lipschitz preserving property of a bounded solution can be directly
shown without the evenness assumption. We refer the reader to Theorem 6.4.4, which
answers a question asked in [98]. A more general question on Lipschitz continuity of viscosity
solutions was posed in [8], but it is not clear if our method here immediately applies to that
general setting.
As for the case of h-convexity preserving property, we obtain the following:
Theorem 1.0.9 (Preservation of right invariant h-convexity, [89]). Assume that f : R2 → R
is Lipschitz. Let u ∈ C(H × [0,∞)) be the unique solution of (1.0.6) with u(·, 0) = u0(·)
satisfying the growth condition (G). Assume in addition that f is concave in R2, i.e.,
f(ξ) + f(η) ≤ 2f
(1
2(ξ + η)
)(1.0.7)
for all ξ, η ∈ R2. If u0 is right invariant h-convex in H; that is,
u0(h−1 · p) + u0(h · p) ≥ 2u0(p)
for all p ∈ H and h ∈ H0, then so is u(·, t) for all t ≥ 0.
Although the result above only concerns the right invariant definitions of Lipschitz con-
tinuity or convexity, we can obtain the left-invariant preserving properties under certain
additional assumptions. For instance, the notions of left invariant and right invariant Lips-
chitz continuity or convexity are equivalent when functions are even or vertically even. This
implies the Lipschitz or h-convexity preserving property of an even function or vertically
even function.
On the other hand, there are also many affirmative examples for Lipschitz and convexity
preserving in the Heisenberg group. In the last section of Chapter 6, we give a list of several
such examples. It is certainly natural to ask whether the preserving of left invariant convexity
holds without these symmetry assumptions. This remains as a future problem we aim to
address.
In Chapter 7, we will study the game-theoretic interpretation of the viscosity solutions
to some fully nonlinear partial differential equations [90]. The discrete game interpretations
8
of various elliptic and parabolic PDEs ([78, 100, 101, 79, 97, 96], etc) have recently attracted
great attention. The game related methods are also used as a new tool in different contexts.
For example, the fattening phenomenon for mean curvature flow is rigorously proved via
games without using parabolic theory by Liu [88]. Armstrong and Smart [7] proved the
uniqueness for infinity harmonic functions using a method related to the tug-of-war games
in [100]. A recent work [92] provides a new proof of Harnack’s inequality for p-Laplacian by
stochastic games. All of these results largely simplify the original PDE proofs and indicate
a strong potential of applicability of the game-theoretic approach.
Due to the works [78, 96, 100, 101], one may find a family of discrete games, whose value
functions uε converge, as ε→ 0, to the unique solution u of a class of quasilinear parabolic
equations including level set mean curvature flow and normalized p-Laplace equations. We
revisit the convexity preserving properties in the Euclidean space for these two classes of
equations by respectively using the game-theoretic approximations first proposed in [96] and
[78]. Our new proofs are based on investigating game strategies or iterated applications of
Dynamic Programming Principles (DPP), which are very different from the standard proofs
in the literature. We look to the convexity preserving property of uε rather than that of
u. For the p-Laplace equations (2 ≤ p ≤ ∞), the convexity of uε follows directly from an
iteration of the corresponding DPP. However, for the level set mean curvature flow equation,
extra work is needed since the control set of the players in the game is not convex. We then
introduce a modified game and manage to show the convexity by comparing the limits of
value functions as ε→ 0. This game-theoretic method can also be applied to study convexity
preserving of the level sets of mean curvature flow equations and the Neumann boundary
problems.
9
2.0 REVIEW OF METRIC MEASURE SPACES
In this chapter, we give a review of some definitions and properties of metric measure spaces.
The abstract metric measure spaces will play a crucial role in the first four chapters. The
main references for this chapter are the books by Ambrosio and Tilli [5], Evans and Gariepy
[33], Heinonen, Koskela, Shanmugalingam and Tyson [58]. Proofs for the theorems without
specifications can be found in the above books.
2.1 BASIC DEFINITIONS IN METRIC MEASURE SPACES
Let X denote a set, and 2X the collection of subsets of X. Let C denote a general constant
whose value can change even in the same chain of estimates.
Definition 2.1.1. A mapping µ : 2X → [0,∞] is called a measure on X if
(1) µ(∅) = 0, and
(2) µ(A) ≤∑∞
k=1 µ(Ak) for any A ⊂⋃∞k=1 Ak.
Definition 2.1.2. A set A ⊂ X is µ-measurable if for every set B ⊂ X,
µ(B) = µ(A ∩B) + µ(B \ A).
Notice that in some textbooks, the mapping µ defined in Definition 2.1.1 is called “outer
measure” while the term “measure” is defined by restricting µ to the collection of all µ-
measurable sets.
A collection of subsets A ⊂ 2X is a σ-algebra if it satisfies ∅, X ∈ A; A ∈ A implies that
X \A ∈ A; Ak ∈ A(k = 1, · · · ) implies that⋃∞k=1 Ak ∈ A. The collection of measurable sets
10
forms a σ-algebra in X. The smallest σ-algebra in X containing all open sets is called the
Borel σ-algebra. The sets in the Borel σ-algebra are called Borel sets.
Let us recall some basic definitions.
(1) A measure µ on X is regular if for every set A ⊂ X, there exists a µ-measurable set B
such that A ⊂ B and µ(A) = µ(B).
(2) A measure µ on X is called Borel if every Borel set is µ-measurable.
(3) A measure µ on X is Borel regular if µ is Borel and for every A ⊂ X, there exists a Borel
set B such that A ⊂ B and µ(A) = µ(B).
(4) A measure µ on X is a Radon measure if µ is Borel regular and µ(K) < ∞ for every
compact set K ⊂ X.
Definition 2.1.3. A function d : X ×X → [0,∞) is called a metric on X if it satisfies
(1) d(x, y) = d(y, x) for all x, y ∈ X;
(2) d(x, y) = 0 if and only if x = y;
(3) d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X.
Then the pair (X, d) is called a metric space. We denote open and closed balls in the
metric space as
B(x, r) = y ∈ X : d(x, y) < r and B(x, r) = y ∈ X : d(x, y) ≤ r.
Sometimes B(x, r) are abbreviated as B. We use 2B to denote a concentric ball of B
with twice the radius and Bn denotes a ball in the Euclidean space Rn.
A metric space (X, d) equipped with a Borel measure µ is called a metric measure space
and denoted as (X, d, µ). In what follows, we always assume that 0 < µ(B) <∞, for every
ball B ⊂ X. We say that µ is doubling if there exists a constant Cd ≥ 1 such that for every
ball B ⊂ X,
µ(2B) ≤ Cdµ(B).
If (X, d, µ) is a doubling metric measure space, then there exists a constant C > 0 such
thatµ(B(x, r))
µ(B(x0, r0))≥ C
( rr0
)s11
whenever x ∈ B(x0, r0), r ≤ r0 and s = logCd/ log 2 is called the associated homogeneous
dimension.
Let µ, ν be Radon measures on a metric space X. Then ν is called absolutely continuous
with respect to µ, written ν << µ, provided µ(A) = 0 implies ν(A) = 0 for all A ⊂ X. The
following theorem is often referred to as Radom-Nikodym theorem.
Theorem 2.1.4. Let µ, ν be Radon measures on a metric space X with ν << µ. Then there
is a µ-measurable function g : X → [0,∞) such that
ν(A) =
ˆA
g dµ
for all µ-measurable sets A ⊂ X.
Given a set A in a metric space (X, d), the diameter of this set is
diam(A) = supd(x, y)|x, y ∈ A.
We next define a very important measure, called Hausdorff measure on a metric space.
Definition 2.1.5. (1) Let A ⊂ X, 0 ≤ s <∞, 0 < δ ≤ ∞. Define
Hsδ(A) = inf
∞∑j=1
α(s)(diamCj
2
)s | A ⊂ ∞⋃j=1
Cj, diamCj ≤ δ,
where
α(s) =πs/2´∞
0e−xxs/2dx
.
(2) For A and s above, define
Hs(A) = limδ→0Hsδ(A) = sup
δ>0Hsδ(A).
We call Hs s-dimensional Hausdorff measure on X.
It can be verified thatHs is a Borel regular measure and n-dimensional Hausdorff measure
Hn coincides with the Lebesgue measure Ln on the Euclidean space Rn.
Definition 2.1.6. The Hausdorff dimension of a set A ⊂ X is defined to be
Hdim(A) = inf0 ≤ s <∞ | Hs(A) = 0.
12
If k > Hdim(A), then Hk(A) = 0, and if k < Hdim(A), then Hk(A) =∞. If k = Hdim(A),
nothing can be said about the value of Hk(A).
Definition 2.1.7. Let (X, d, µ) be a metric space equipped with a Borel measure µ and let
s > 0. We say that a metric measure space (X, d, µ) is s-Ahlfors regular if there is a constant
CA ≥ 1, such that
C−1A rs ≤ µ(B(x, r)) ≤ CAr
s, for any x ∈ X and 0 < r < diam(X).
If X is s-Ahlfors regular with respect to a Borel regular measure µ, then X has Hausdorff
dimension precisely s. Moreover, µ is comparable to the Hausdorff measure Hs, that is, there
exist constant C ≥ 1 such that
C−1Hs(E) ≤ µ(E) ≤ CHs(E),
for all Borel sets E ⊂ X. This implies that X is s-Ahlfors regular with respect to Hs [55,
Exercise 8.11.].
Definition 2.1.8. A metric space (X, d) is uniformly perfect if there exists a constant
0 < c < 1 such that for all x ∈ X and 0 < r < diam(X),
B(x, r) \B(x, cr) 6= ∅.
It is well known that s-Ahlfors regular metric spaces are uniformly perfect. Indeed, if X
is s-Ahlfors regular with constant CA, then B(x, r) \ B(x, cr) 6= ∅ for any 0 < c < C−2/sA .
Suppose to the contrary that B(x, r) \B(x, cr) = ∅. This would imply
C−1A rs ≤ Hs(B(x, r)) = Hs(B(x, cr)) ≤ CA(cr)s
so C−2/sA ≤ c, which is a contradiction. This is observed in some earlier papers, for example
[114].
Let f : X → R be a function on a metric space. If there exists a constant L ≥ 0 such
that
|f(x)− f(y)| ≤ Ld(x, y)
13
for all x, y ∈ X, we say that f is a Lipschitz function on X. The smallest constant L with
this property will be denoted by ‖f‖Lip. For α ∈ (0, 1), if there exists a constant L ≥ 0 such
that
|f(x)− f(y)| ≤ Ld(x, y)α
for all x, y ∈ X, we say that f is an α-Holder continuous function.
2.2 CURVES IN METRIC MEASURE SPACES
A curve in a metric space (X, d) is a continuous map from an interval into X. We usually
denote a curve as γ : [a, b]→ X, and the image of a curve γ([a, b]) as Γ.
We define the length of a curve γ : [a, b]→ X as
`(γ) = supk∑i=1
d(γ(ti), γ(ti−1)), a = t0 < t1 < · · · < tk = b,
where the supremum is taken over all finite sequences of ti ⊂ [a, b] defined as above. If
`(γ) < ∞, then we say the curve γ is a rectifiable curve. Every rectifiable curve admits
a parametrization by arc-length, that is, there exists a parametrization of Γ, denoted as
γ : [0, `(γ)]→ X such that `(γ|[0,t]) = t for any t ∈ [0, `(γ)]. This arc-length parametrization
γ is a 1-Lipschitz mapping.
Definition 2.2.1. For a curve γ : [a, b] → X we define speed at a point t ∈ (a, b) as the
limit
|γ′|(t) = limh→0
d(γ(t+ h), γ(t))
|h|,
if the limit exists.
Theorem 2.2.2. For every Lipschitz curve γ : [a, b] → X speed exists almost everywhere
and
`(γ) =
ˆ b
a
|γ′|(t) dt.
14
If a rectifiable curve is parametrized by arc-length γ, then |γ′(t)| = 1 for almost every
point t ∈ [0, `(γ)].
We say a metric space X is quasiconvex if there is a constant C ≥ 1 such that every
two points x, y ∈ X can be joined by a curve with length less than or equal to Cd(x, y). A
metric space is called proper if bounded and closed sets are compact. The following theorem
guarantees the existence of geodesics in proper metric spaces.
Theorem 2.2.3. Suppose a metric space X is proper and path connected, that is, every
two points x, y can be joined by a rectifiable curve in X. Then there exists a shortest curve
γ : [a, b] → X connecting x and y, that is, γ(a) = x and γ(b) = y. The shortest curve is
injective.
Next, we define the integral along a rectifiable curve.
Definition 2.2.4. Let γ : [a, b] → X be a rectifiable curve and g : X → [0,∞] be a Borel
measurable function. Then we define
ˆγ
g =
ˆ `(γ)
0
g(γ(t)) dt,
where γ is the arc-length parametrization of the given curve.
We also give the generalization of the Euclidean area formula to the case of Lipschitz
maps f from the Euclidean space Rn into a metric space X. The proof can be found in [77,
Corollary 8].
Theorem 2.2.5 (Area fomula). Let f : Rn → X be Lipschitz. Then
ˆRn
θ(x)Jn(mdfx) dx =
ˆX
∑x∈f−1(y)
θ(x) dHn(y)
for any Borel function θ : Rn → [0,∞] and
ˆA
θ(f(x))Jn(mdfx) dx =
ˆX
θ(y)H0(A ∩ f−1(y)) dHn(y)
for A ∈ B(Rn) and any Borel function θ : X → [0,∞].
15
Here Jn(mdfx) in the Jacobian of the metric derivative. We will however, be interested in
a special case where the Lipschitz mapping into X is just an injective arc-length parametrized
rectifiable curve γ : [0, `] → X. In this case, J1(mdfx) in the above theorem is the metric
derivative |γ′|(t) and hence it equals 1. Let Γ = γ([0, `]) and g : X → [0,∞] be a Borel
function. Applying the area formula, we get
ˆ `
0
g(γ(s)) ds =
ˆΓ
g(y) dH1(y).
If N ⊂ X and H1(N) = 0, the area formula also implies that H1(γ−1(N)) = 0.
2.3 RECTIFIABILITY OF ONE-DIMENSIONAL COMPACT AND
CONNECTED SPACES
We list here several important results about the parametrization of compact and connected
metric measure spaces with finite H1 measure.
The first result proved by Schul [110, Lemma 2.3] gives a Lipschitz parametrization of
such spaces.
Lemma 2.3.1. Let K ⊂ X be a compact connected set of finite H1 measure. Then there is a
Lipschitz function γ : [0, 1]→ K such that γ([0, 1]) = K and ‖γ‖Lip ≤ 32H1(K). Moreover,
if K is 1-Ahlfors-regular, then
R
C≤ H1(γ−1(B(x,R))) ≤ CR ∀x ∈ K, 0 < R ≤ diam(K), (2.3.1)
where C is a constant depending only on the 1-Ahlfors regularity constant of the set K.
The proofs of the following two theorems can be found in [5, Theorem 4.4.7, Theorem
4.4.8].
Theorem 2.3.2 (First rectifiability theorem). If X is a compact and connected set and
H1(X) < ∞, then every pair of points x, y ∈ X can be connected by a injective rectifiable
curve.
16
We call curves without self-intersections simple curves, that is, there is an injective
parametrization of such curves.
Theorem 2.3.3 (Second rectifiability theorem). If X is compact, connected, and H1(X) <
∞, then there exist countably many arc-length parametrized simple curves γi : [0, `i] → X
such that
H1(X \∞⋃i=1
γi([0, `i])) = 0.
We briefly describe the construction of this parametrization. Since X is compact, we can
choose x, y ∈ X such that
d(x, y) = diam(X).
By Theorem 2.3.2, we can join x, y by an arc-length parametrized simple curve γ0 : [0, `0]→
X and we denote the range of this curve as Γ0. Suppose that we have already constructed
Γ0, · · · ,Γk with the following properties:
(1) Γi ⊂ X, i = 0, · · · , k;
(3) Each curve γi : [0, `i]→ X with Γi = γi([0, `i]) is a simple arc-length parametrized curve;
(2) Each intersection Γi ∩⋃j<i Γj consists of a single point, for i = 1, · · · , k.
Let
dk = supx∈X
d(x,
k⋃i=0
Γi).
If dk = 0, then X =⋃ki=0 Γi and we are done. If dk > 0 for all k, by compactness we
can choose xk ∈ X and yk ∈ ∪ki=0Γi such that d(xk, yk) = dk. Connect xk and yk with an
arc-length parametrized simple curve γk+1 such that γk+1(0) = xk and γk+1(`k+1) = yk+1.
Let
t = inft ∈ [0, `k+1]| γk+1(t) ∈k⋃i=0
Γi
and define Γk+1 = γk+1([0, t]). We get that Γk+1 ⊂ X is an arc-length parametrized simple
curve and the intersection of Γk+1 and⋃ki=0 Γi consists of one single point. We can continue
this construction. Since⋃∞i=0 Γi may not be closed, we may have
X \∞⋃i=0
Γi 6= ∅.
17
We omit the arguments to estimate the above set X \⋃∞i=0 Γi has 1-dimensional Hausdorff
measure zero.
It is easy to see from this construction that these simple curves intersect with each other
at most at one point. If two curves intersect with each other, then the intersection point
must be the endpoint of one of these curves.
18
3.0 SOBOLEV FUNCTIONS ON METRIC MEASURE SPACES
The theory of Sobolev functions has been widely applied in different areas of mathematics
including calculus of variations, partial differential equations, and so on. In this chapter,
we will first give a brief review of the classical theory of Sobolev functions. We refer to the
book by Evans and Gariepy [33] and notes by Haj lasz [48] for the definitions and proofs of
the theorems.
There are various extensions of the classical theory of Sobolev functions to general metric
measure spaces and their connections with variational problems, geometric function theory
and many related fields have been studied since the nineties. These results constitute a
significant part of analysis on metric measure spaces. In this chapter, we list several impor-
tant definitions of Sobolev functions on metric measure spaces and also give the definition
of spaces supporting Poincare inequalities in the end. Doubling metric measure spaces that
support Poincare inequalities provide a setting in which different definitions of Sobolev spaces
are equivalent. The main source for these materials are the survey paper by Haj lasz [47] and
the book by Heinonen, Koskela, Shanmugalingam and Tyson [58].
3.1 SOBOLEV FUNCTIONS IN THE EUCLIDEAN SPACE
3.1.1 Definitions and basic properties
In this section Ω will denote an open subset of Rn, C∞c (Ω) be the collection of all smooth
functions with compact support in Ω and C∞(Ω) be the collection of all smooth functions
in Ω.
19
Definition 3.1.1. Let u ∈ L1loc(Ω) and 1 ≤ i ≤ n. We say gi ∈ L1
loc(Ω) is the weak partial
derivative of u with respect to xi if
ˆΩ
u∂ϕ
∂xi= −
ˆΩ
giϕ
for all ϕ ∈ C∞c (Ω).
It is easy to verify that given a function u ∈ L1loc(Ω), if the weak partial derivative exists,
then it is uniquely defined Ln almost everywhere. We write
∂u
∂xi= gi
and
∇u =( ∂u∂x1
, . . . ,∂u
∂xn
)Definition 3.1.2. For 1 ≤ p < ∞, the function u belong to the Sobolev space W 1,p(Ω) if
u ∈ Lp(Ω) and the weak partial derivatives ∂u/∂xi exist and belong to Lp(Ω) for 1 ≤ i ≤ n.
If u ∈ W 1,p(Ω), define
‖u‖W 1,p(Ω) = ‖u‖Lp(Ω) + ‖∇u‖Lp(Ω),
and we can verify that the above definition is a norm and Sobolev spaces equipped with this
norm is a Banach space.
Theorem 3.1.3 (Meyers-Serrin). The smooth functions C∞(Ω) are dense in W 1,p(Ω).
According to this theorem, the Sobolev functions can also be defined as the completion
of all smooth functions with respect to the Sobolev norm. Moreover, if Ω = Rn, C∞c (Rn) is
dense in W 1,p(Rn).
The Sobolev embedding theorems are among the most important results in the theory
of Sobolev spaces. We list some results below.
20
Theorem 3.1.4 (Gagliardo-Nirenberg-Sobolev inequality). Let 1 ≤ p < n, we define
p∗ =np
n− p,
and call p∗ the Sobolev conjugate of p. Then for all u ∈ W 1,p(Rn), there exists a constant
C(p, n), such that
‖u‖Lp∗(Rn) ≤ C‖∇u‖Lp(Rn).
Theorem 3.1.5. For each n < p <∞, there exists a constant C(p, n) such that
|u(x)− u(y)| ≤ C|x− y|1−np ‖∇u‖Lp(B)
for all x, y ∈ B and u ∈ W 1,p(B).
As a corollary, if u ∈ W 1,p(Ω) and n < p < ∞ then u is locally Holder continuous with
exponent 1−n/p. However, when 1 ≤ p < n, Sobolev functions need not be continuous. For
example, u(x) = 1/|x|1/2 ∈ W 1,1(B2(0, 1)) and it is discontinuous in the origin.
The above examples and theorems show that in Rn, the properties of Sobolev functions
W 1,p are very different when p > n and p < n. We often call Sobolev functions in W 1,p with
p = n the Sobolev functions in the critical case. The embedding result in the critical case is
as follows.
Theorem 3.1.6 (Trudinger). Let Ω ⊂ Rn be a bounded Lipschitz domain. Then there exist
constants C1, C2 depending on Ω only such that
Ω
exp( |u− uΩ|C1‖∇u‖Ln(Ω)
) nn−1 ≤ C2.
for any u ∈ W 1,n(Ω).
21
Here and in what follows we will use notation
uΩ =
Ω
u dµ =1
µ(Ω)
ˆΩ
u dµ.
When p = n and n ≥ 2, u(x) = log | log |x|| provides an example of a discontinuous
function in W 1,n(Bn(0, 1)). When n = 1, functions in W 1,1([a, b]) are absolutely continuous.
A function u : [a, b]→ R is absolutely continuous if for every ε > 0, there exists a δ > 0 such
that
k∑i=1
|u(bi)− u(ai)| < ε,
for any finite collection of pairwise disjoint subintervals⋃ki=1[ai, bi] ⊂ [a, b] satisfying
k∑i=1
|bi − ai| < δ.
We denote the collection of absolutely continuous functions on [a, b] as AC([a, b]). An abso-
lutely continuous function u is differentiable almost everywhere and we denote this pointwise
derivative as u′. In fact, u ∈ AC([a, b]) if and only if there exist g ∈ L1([a, b]) such that
u(x) = u(a) +
ˆ x
a
g(t) dt.
Moreover, u′(t) = g(t) almost everywhere. In other words absolutely continuous functions
are exactly the functions for which the fundamental theorem of calculus is true.
We denote the collection of all absolutely continuous functions with u′ ∈ Lp([a, b]) as
ACp([a, b]). Upon choosing a representative, we have ACp([a, b]) = W 1,p([a, b]).
The following result is the classical Poincare inequality.
Theorem 3.1.7. If u ∈ W 1,p(B(r)) and 1 ≤ p <∞, then
( B
|u− uB|p dx)1/p ≤ C(n, p)r
( B
|∇u|p dx)1/p
.
22
3.1.2 Equivalent characterizations
In this section, we give several equivalent definitions of Sobolev spaces in Rn. In the previous
section, we have seen the equivalence of Sobolev functions in W 1,p([a, b]) with absolutely
continuous functions in ACp([a, b]). This characterization has a higher dimensional analog.
If Ω ⊂ R is open, we say that u ∈ AC(Ω) if u is absolutely continuous on every compact
interval in Ω.
Definition 3.1.8. Let Ω ⊂ Rn be an open set. We say that u is absolutely continuou on
lines, u ∈ ACL(Ω), if the function u is Borel measurable and for almost every line ` parallel
to one of the coordinate axes, u|` ∈ AC(Ω ∩ `).
Since absolutely continuous functions in the real line are differentiable almost everywhere,
u ∈ ACL(Ω) has partial derivative almost everywhere and hence ∇u is defined almost
everywhere. We say that u ∈ ACLp(Ω) if u ∈ Lp(Ω) ∩ ACL(Ω) and |∇u| ∈ Lp(Ω).
Theorem 3.1.9. For 1 ≤ p <∞ and any open set Ω ⊂ Rn,
W 1,p(Ω) = ACLp(Ω).
Moreover, the pointwise partial derivatives of an ACLp(Ω) function equal the weak partial
derivatives.
Let M be the Hardy-Littlewood maximal function defined as
Mu(x) = supr>0
B(x,r)
|u(y)| dy
and
MRu(x) = sup0<r<R
B(x,r)
|u(y)| dy
for x ∈ Rn.
Theorem 3.1.10. For u ∈ Lp(Rn), 1 < p <∞, the following conditions are equivalent
(1) u ∈ W 1,p(Rn),
23
(2) There exists 0 ≤ g ∈ Lp(Rn) and σ ≥ 1 such that
B
|u− uB| dx ≤ r(
σB
gp dx)1/p
on every ball B of any radius r.
(3) There exists 0 ≤ g ∈ Lp(Rn) and σ ≥ 1 such that
|u(x)− u(y)| ≤ |x− y|(g(x) + +g(y)
)almost everywhere.
The second condition follows from the Poincare inequality 3.2.1 and Holder inequality. It
is first proved by Koskela and MacManus [82]. The third condition for u ∈ W 1,p(Rn) stems
from the following inequality
|u(x)− u(y)| ≤ |x− y|((M|∇u|p(x))1/p + (M|∇u|p(y))1/p
)a.e..
This characterization is first proved by Haj lasz [46].
3.2 SOBOLEV FUNCTIONS ON THE METRIC MEASURE SPACES
Note that the conditions (2) and (3) in Theorem 3.1.10 do not involve derivatives and only
involve metric and measure in Rn. Each of this condition can be used to define a version of
Sobolev spaces in general metric measure spaces and will be discussed in Sections 3.2.1 and
3.2.2. However, in Theorem 3.1.9, the notion of almost all lines parallel to coordinate axes
and the notion of gradient do not apply to the general metric measure spaces. Instead, we will
introduce the modulus of the path family and the notion of upper gradient as replacements.
The Sobolev space based on this approach is called Newton Sobolev space and will be
discussed in Section 3.2.3.
24
3.2.1 Sobolev spaces M1,p
Definition 3.2.1. Let (X, d, µ) be a metric space equipped with a Borel measure µ. For
0 < p < ∞ we define the Haj lasz Sobolev space M1,p(X, d, µ) to be the set of all functions
u ∈ Lp(X) for which there exists a nonnegative Borel function g ≥ 0 such that
|u(x)− u(y)| ≤ d(x, y)(g(x) + g(y)
)almost everywhere.
Denote by D(u) the class of all nonnegative Borel functions g that satisfy the above in-
equality. Thus u ∈ M1,p(X, d, µ) if and only if u ∈ Lp(X) and D(u) ∩ Lp 6= ∅. The space
M1,p(X, d, µ) is linear and we define
||u||M1,p = ||u||Lp + infg∈D(u)
||g||Lp .
When p ≥ 1, || · ||M1,p is a norm and M1,p(X) is a Banach space [46]. For studies and
applications of Haj lasz Sobolev spaces with 0 < p < 1, see for example [47, 68, 83, 84, 85].
According to Theorem 3.1.10, in the Euclidean space and 1 < p <∞,
M1,p(Rn, | · |,Ln) = W 1,p(Rn, | · |,Ln).
The equivalence of M1,p and W 1,p also holds on smooth domains Ω in Rn. However, in
general, M1,1 ( W 1,1. For example, function u(x) = −x/(|x| log |x|) ∈ W 1,1(I) but it does
not belong to M1,1(I), where I = (−1/4, 1/4).
An analog of the Sobolev embedding theorems holds for functions in M1,p(X, d, µ), where
X is a doubling space with constant Cd and the associated homogeneous dimension s =
logCd/ log 2 plays the role of dimension in the Euclidean space. The following embedding
theorem is proved by Haj lasz [46].
Theorem 3.2.2. Let X be a doubling space and fix a ball B ⊂ X of radius r, σ > 1. Assume
that u ∈ M1,p(σB, d, µ) and g ∈ D(u) ∩ Lp(X), where 0 < p < ∞. There exist constants
C,C1 and C2 depending on Cd, p and σ only such that
(1) If 0 < p < s, then u ∈ Lp∗(B), p∗ = sp/(s− p), and
infc∈R
( B
|u− c|p∗ dµ)1/p∗ ≤ Cr
( σB
gp dµ)1/p
.
25
(2) If p = s, then B
exp(C1µ(σB)1/s
r
|u− uB|‖g‖sL(σB)
)dµ ≤ C2.
(3) If p > s, then u is Holder continuous on B and
|u(x)− u(y)| ≤ Crs/pd(x, y)1−s/p( σB
gp dµ)1/p
for x, y ∈ B.
3.2.2 Sobolev spaces P 1,p
Definition 3.2.3. Fix σ ≥ 1 and 0 < p < ∞. We say that a pair (u, g), u ∈ L1loc(X),
0 ≤ g ∈ Lploc(X) satisfies the p-Poincare inequality if the following inequality holds:
B
|u− uB|dµ ≤ r
( σB
gpdµ
) 1p
(3.2.1)
on every ball B of radius r and σB ⊂ X.
The class of u ∈ Lp(X) for which there exists 0 ≤ g ∈ Lp(X) so that the pair (u, g)
satisfies the p-Poincare inequality will be denoted by P 1,pσ (X, d, µ) and
P 1,p(X, d, µ) =⋃σ≥1
P 1,pσ (X, d, µ).
By Theorem 3.1.10, the Sobolev space P 1,p defined above is equivalent with the classical
Sobolev space in the Euclidean setting.
Theorem 3.2.4. Let (X, d, µ) be a doubling metric measure space and s = log2Cd is the
associated homogeneous dimension. If p ≥ s/(s+ 1), then
M1,p(X) ⊂ P 1,p(X).
The above theorem is proved by Haj lasz [47]. We also give the general Sobolev embedding
theorem for P 1,p(X). This result is due to Haj lasz and Koskela [49].
Theorem 3.2.5. Let (X, d, µ) be a doubling metric measure space and s = log2Cd be the
associated homogeneous dimension. Assume the pair u ∈ Lp(X) and 0 ≤ g ∈ Lp(X) satisfies
the p-Poincare inequality (3.2.1) with 0 < p <∞ and σ > 1.
26
(1) If 0 < p < s, then for every 0 < h < p∗ = sp/(s− p)
infc∈R
( B
|u− c|hdµ)1/h ≤ Cr
( 6σB
gp dµ)1/p
.
If in addition g ∈ Lq, p < q < s, then
infc∈R
( B
|u− c|q∗dµ)1/q∗ ≤ Cr
( 6σB
gq dµ)1/q
,
where q∗ = sq/(s− q) and B is any ball of radius r.
(2) If p = s, then ˆB
exp(C1µ(6σB)1/s|u− uB|
r‖g‖Ls(6σB)
)dµ ≤ C2.
(3) If p > s, then u is locally Holder continuous and
|u(x)− u(y)| ≤ Crs/pd(x, y)1−s/p( 6σB0gp
dµ)1/p
for x, y ∈ B, where B is an arbitrary ball of radius r0.
The constants in the theorem depend on p, q, h, Cd and σ.
3.2.3 Sobolev spaces N1,p
Definition 3.2.6. Let Γ be a collection of non constant rectifiable curves, and let F (Γ) be
the family of all Borel measurable functions ρ : X → [0,∞] such that
ˆγ
ρ ≥ 1 for every γ ∈ Γ.
For each 1 ≤ p <∞ we define
Modp(Γ) = infρ∈F(Γ)
ˆX
ρpdµ.
The number Modp(Γ) is called the p-modulus of the family Γ.
We can verify that Modp is a measure on the family of all nonconstant rectifiable curves
in X. If some property holds for all nonconstant rectifiable curves except a subcollection Γ
with Modp(Γ) = 0, then we say that the property holds for p-a.e. curve. By the following
theorem, we can see that this notion of p-a.e. curve is a natural generalization of almost
every line parallel to a given coordinate axe.
27
Theorem 3.2.7. Points in Qn = [0, 1]n = [0, 1]×Qn−1 will be denoted by x = (x1, x′). Let
1 ≤ p < ∞. For a Borel subset E ⊂ Qn−1, consider the family of straight segments passing
through E and parallel to x1, i.e.,
ΓE = γx′ : [0, 1]→ Qn : γx′ = (t, x′), x′ ∈ E.
Then Modp(ΓE) = 0 if and only if Ln(E) = 0.
Definition 3.2.8. Let u : X → R be a Borel function. We say that a Borel function
g : X → [0,∞] is an upper gradient of u if
|u(γ(a))− u(γ(b))| ≤ˆγ
g
for every rectifiable curve γ : [a, b] → X. We say that g is a p-weak upper gradient of u if
the above inequality holds for p-a.e. curve.
If g is a p-weak upper gradient of u which is finite a.e., then for every ε > 0, there is an
upper gradient gε of u such that
gε ≥ g everywhere, and ‖gε − g‖Lp < ε.
This shows that a p-weak upper gradient of u can be nicely approximated by an upper
gradient.
Moreover, the notion of upper gradient is a natural generalization of the length of the
gradient. If u ∈ C∞(Ω) and Ω ⊂ Rn, then we can verify that |∇u| is an upper gradient of
u. Furthermore, |∇u| is the least upper gradient in the sense that if g ∈ L1loc(Ω) is another
upper gradient of u, then g ≥ |∇u| a.e.. In fact, we have a stronger result.
Theorem 3.2.9. Any function u ∈ W 1,p(Ω), 1 ≤ p <∞ has a representative for which |∇u|
is a p-weak upper gradient. On the other hand, if g ∈ L1loc(Ω) is a p-weak upper gradient of
u, then g ≥ |∇u| a.e..
28
Let 1 ≤ p < ∞. N1,p(X, d, µ) is the class of all Lp-integrable Borel functions on X for
which there exists a p-integrable p-weak upper gradient. With each u ∈ N1,p(X, d, µ) we
associate a seminorm
‖u‖N1,p(X,d,µ) = ‖u‖Lp + infg‖g‖Lp ,
where the infimum is taken over all p-weak upper gradients g of u.
Definition 3.2.10. We define an equivalence relation in N1,p(X, d, µ) by u ∼ v if
‖u− v‖N1,p(X,d,µ) = 0.
Then the space N1,p(X, d, µ) is defined as the quotient space N1,p(X, d, µ)/ ∼ and it is
equipped with the norm
‖u‖N1,p = ‖u‖N1,p .
This space N1,p(X, d, µ) with the above norm is a Banach space for 1 ≤ p <∞ and it is
a natural generalization of the classical Sobolev space W 1,p to the setting of metric spaces.
Theorem 3.2.11. If Ω ⊂ Rn is open and 1 ≤ p <∞, then
N1,p(Ω, | · |,Ln) = W 1,p(Ω, | · |,Ln)
as sets and the norms are equal.
3.3 SPACES SUPPORTING THE POINCARE INEQUALITY
The notion of an abstract Poincare inequality on metric measure spaces was introduced by
Heinonen and Koskela [56]. Metric measure spaces that are doubling and support an abstract
Poincare inequality provide a good structure to study the first-order analysis.
29
Definition 3.3.1. Let p ≥ 1. A metric measure space (X, d, µ) is said to support a p-
Poincare inequality if there exists constants C, λ ≥ 1 such that for all measurable functions,
the following holds: B
|u− uB|dµ ≤ CdiamB
( λB
gpdµ
) 1p
,
for every pair of functions u : X → R and g → [0,∞] where u is measurable, and g is an
upper gradient for u.
For examples of spaces supporting Poincare inequalities, we give a partial list below:
Euclidean spaces, compact Riemannian manifolds, complete Riemannian manifolds with
nonnegative Ricci curvature [19, 108], Carnot groups and general sub-Riemannian manifolds
equipped with Carnot-Caratheodory metrics [49, 54, 57], boundaries of hyperbolic buildings
[18], metric spaces with quantitative topology [111] and the Laakso spaces [86]. If X is
complete, doubling and supports a p-Poincare inequality for p ≥ 1, then X is quasiconvex
[24, Theorem 17.1.][49, Proposition 4.4]. In this manuscript, we will prove that each compact,
quasiconvex and 1-Ahlfors regular space also supports p-Poincare inequality for 1 ≤ p <∞.
As mentioned before, a doubling metric measure spaces that supports Poincare inequal-
ities provides a very good setting such that different definitions of Sobolev spaces are equiv-
alent. The proof of the following theorem can be found in [47, Theorem 11.3]
Theorem 3.3.2. Let (X, d, µ) be a complete metric measure space with µ Borel and doubling.
If 1 < p <∞ and the space supports the q-Poincare inequality for some 1 ≤ q < p, then
M1,p(X, d, µ) = N1,p(X, d, µ) = P 1,p(X, d, µ).
By Holder’s inequality, we know that any metric measure space that supports a (1, p)-
Poincare inequality also supports a (1, q)-Poincare inequality for 1 ≤ p ≤ q. Keith and
Zhong [74, Theorem 1.0.1] proved that the parameter p > 1 in the space supporting Poincare
inequality is also open ended on the left in a doubling metric space. We list their result as
below.
Theorem 3.3.3. Let p > 1 and (X, d, µ) be a complete metric measure space with µ Borel
and doubling, that admits a (1, p)-Poincare inequality. Then there exists ε > 0 such that
(X, d, µ) admits a (1, q)-Poincare inequality for every q > p− ε, quantitatively.
30
Combining the above two theorems, we have the following corollary.
Corollary 3.3.4. Let (X, d, µ) be a complete metric measure space with µ Borel and dou-
bling. If 1 < p <∞ and the space supports the p-Poincare inequality, then
M1,p(X, d, µ) = N1,p(X, d, µ) = P 1,p(X, d, µ).
31
4.0 SOBOLEV FUNCTIONS IN THE CRITICAL CASE ON METRIC
MEASURE SPACES
Let Ω ⊂ Rn be a domain. We define the critical exponent of the Sobolev space W 1,p(Ω) to
be p = n. This is because that when p > n, W 1,p functions are Holder continuous while for
p < n they need not be continuous at any point. This notion can be extended to the settings
of metric measure spaces. If (X, d, µ) is a doubling metric measure space then we define
the critical exponent for the Sobolev space M1,p(X), P 1,p(X) or N1,p(X) to be equal to the
homogeneous dimension p = s. This definition is suggested by the corresponding Sobolev
embedding Theorems 3.2.2 and 3.2.5.
From the examples in the Euclidean spaces, we conjecture that the Sobolev functions in
the critical case may be discontinuous if the homogeneous dimension satisfies s > 1. Thus
we turn to investigate the continuity of Sobolev functions on metric measure spaces with
dimensions less than or equal to 1. One natural generalization is to compact, connected 1-
dimensional metric spaces. We give the definition of the absolutely continuous functions on
such spaces and prove the equivalence of Newton Sobolev functions in N1,1(X) and absolutely
continuous functions. Moreover, by proving that the spaces support Poincare inequality, we
get the equivalence of these absolutely continuous functions with Sobolev functions defined
via different approaches.
Another direction is to look at metric spaces with dimension s ≤ 1. We focus on the
case of s-Ahlfors regular metric spaces which include a large class of fractals and we prove
the uniform continuity of Haj lasz Sobolev functions with explicit estimates.
32
4.1 ON COMPACT, CONNECTED METRIC SPACES WITH H1(X) <∞
4.1.1 Absolutely continuous functions on the metric spaces
We define absolutely continuous functions on spaces of finite 1-dimensional measure as fol-
lows.
Definition 4.1.1. Let (X, d,H1) be a compact, connected metric space with H1(X) < ∞.
A function u : X → R is absolutely continuous if for any ε > 0, there is a positive number δ
such that ∑i
|u γi(`i)− u γi(0)| < ε,
for any collection of pairwise disjoint arc-length parametrized simple curves γi : [0, `i] → X
with total length∑
i `i < δ.
Let V (f, [a, b]) denote the total variation of a mapping f : [a, b]→ X on [a, b]. The total
variation of a mapping on a interval is defined as
V (f, [a, b]) := sup
k∑i=1
d(f(ti), f(ti−1)), a = t0 < t1 < · · · < tk = b
,
where the supremum is taken over all finite sequences of ti ⊂ [a, b] defined as above. We
can replace |u γi(`i) − u γi(0)| by V (u γi, [0, `i]) in the above definition, and get an
equivalent definition.
We denote the above class of absolutely continuous functions on X by AC(X). Let
X = [a, b], it is easy to verify this definition is consistent with the classical definition.
We prove that the absolutely continuous functions defined above are uniformly continuous
on a quasiconvex metric space.
Proposition 4.1.2. Let (X, d, µ) be a compact, quasiconvex metric measure space with
H1(X) < ∞. If a function u : X → R is absolutely continuous, then it is uniformly
continuous.
Proof. Given an arbitrary positive number ε > 0, it suffices to find δ > 0 such that |u(x)−
u(y)| < ε whenever d(x, y) < δ for x, y ∈ X.
33
Since X is compact and path-connected, two arbitrary points x, y ∈ X can be joined
by a shortest curve. We denote the arc-length parametrization of this shortest curve by
γ0 : [0, `0]→ X with γ0(0) = x and γ0(`0) = y. Thus,
|u(x)− u(y)| = |u(γ0(0))− u(γ0(`0))| = |u γ0(0)− u γ0(`0)|.
The fact that the space is quasiconvex means that there is a curve connecting x, y with
length ` ≤ Cd(x, y). Thus, the length of the shortest curve `0 ≤ ` ≤ Cd(x, y).
Since u ∈ AC(X), for an arbitrary positive number ε > 0, there is δ0 > 0, such that if
the length of a simple curve `0 < δ0, then
|u γ0(`0)− u γ0(0)| < ε.
In particular, let δ < C−1δ0, then `0 ≤ Cd(x, y) < δ0. It implies that
|u(x)− u(y)| < ε.
4.1.2 Absolutely continuous characterization of Sobolev functions
In this section, we will show that the absolutely continuous functions we define are actually
the same as Newton Sobolev functions.
Theorem 4.1.3. Let (X, d,H1) be a compact, connected metric measure space with H1(X) <
∞. If u ∈ AC(X), then there is an upper gradient g ∈ L1(X) for u, that is, for any rectifiable
curve γ : [a, b]→ X, we have
|u(γ(a))− u(γ(b))| ≤ˆγ
g.
34
Proof. By Theorem 2.3.3, we know that there is a countable collection of simple curves in
X and a set N1 with H1(N1) = 0 such that
X =∞⋃i=1
γi((0, `i)) ∪N1.
Let Γi = γi([0, `i]), then the intersection of Γi are Γj is empty for i 6= j. Indeed, the simple
curves from Theorem 2.3.3 meet at the endpoints only as we excluded the endpoints here.
We denote u γi by ui. If u ∈ AC(X), then ui ∈ AC((0, `i)). Thus, u′i(t) exists almost
everywhere for t ∈ (0, `i). Moreover, we have
ui(`i)− ui(0) =
ˆ `i
0
u′i(s) ds,
and`i∨0
ui =
ˆ `i
0
|u′i(s)| ds.
We denote the collection of points x ∈ X such that u′i(γ−1i (x)) does not exist as N2, and
N0 = N1 ∪N2. It is clear that H1(N0) = 0. Then we define a function g : X → R as follows,
g(x) =
|u′i(γ−1i (x))| if x ∈ X \N0
∞ if x ∈ N0.
(4.1.1)
This function g is integrable on X. In fact, by definition and area formula, we have
ˆX
g dH1 =
ˆX\N0
g dH1
=∑i
ˆΓi
g dH1
=∑i
ˆ `i
0
g(γi(s)) ds
=∑i
ˆ `i
0
|u′i(s)| ds
=∑i
`i∨0
ui.
35
Since γi is injective and H1(X) <∞, it implies that
∑i
`γi =∑i
H1(Γi) ≤ H1(X) <∞.
Thus, for any ε > 0, there exists a natural number n0 such that∑∞
i=n0+1 `γi < ε. By
Definition 4.1.1, it implies that∑∞
i=n0+1
∨`i0 ui can be sufficiently small.
ˆX
g dH1 =∑i
`i∨0
ui
=
n0∑i=1
`i∨0
ui +∞∑
i=n0+1
`i∨0
ui <∞.
We next prove that g is an upper gradient for the function u ∈ AC(X), that is, for any
rectifiable curve γ : [a, b]→ X, we have
|u(γ(a))− u(γ(b))| ≤ˆγ
g.
Let γ : [a, b]→ X be a rectifiable curve and Γ = γ([a, b]). Since Γ is compact, connected
and H1(Γ) < ∞, there exists a shortest curve joining γ(a) and γ(b) in Γ. We denote the
arc-length parametrization of this injective curve by γ0 : [0, `]→ Γ with
γ0(0) = γ(a) and γ0(`) = γ(b). (4.1.2)
Let Γ0 = γ0([0, `]). Applying the area formula we obtain that
ˆγ0
g =
ˆ `
0
g(γ0(s)) ds
=
ˆΓ0
g(x) dH1
≤ˆ
Γ
g(x) dH1
≤ˆγ
g.
(4.1.3)
Since u ∈ AC(X), it follows that u0 = uγ0 ∈ AC([0, `]) and u′0 exists almost everywhere.
Let I = i ∈ N : H1(Γ0 ∩ Γi) 6= 0. Then Γ0 is the union of⋃i∈I(Γ0 ∩ Γi) and a null set.
36
Let i ∈ I and x ∈ Γi ∩ Γ0 such that u′0(γ−10 (x)), u′i(γ
−1i (x)) both exist and t = γ−1
0 (x) is
a density point of γ−10 (Γi ∩ Γ0). By definition,
|u′0(t)| = limh→0
|u0(t+ h)− u0(t)||h|
= limh→0
|u(γ0(t+ h))− u(γ0(t))||h|
.
Since t is a density point in the measurable set γ−10 (Γi ∩ Γ0), when h is sufficiently small,
there exists t + h ∈ γ−10 (Γi ∩ Γ0). We denote γi(s) = γ0(t) and γi(s
′) = γ0(t + h). By
construction, the curve γi is the shortest curve joining γi(s) and γi(s′). Since γ0 and γi are
both parametrized by arc-length, it follows that
|s− s′| = `γi(s,s′) ≤ `γ0(t,t+h) = |h|.
It follows that
|u′0(t)| = limh→0
|u(γ0(t+ h))− u(γ0(t))||h|
≤ lim|s−s′|→0
|u(γi(s′))− u(γi(s))||s− s′|
= |u′i(s)|.
By area formula, for the injective arc-length parametrized curves γi and γ0, the H1 measure
of the preimage of a null set is zero. Thus, it implies that
|u′0(t)| ≤ |u′i(s)| = |u′i(γ−1i (γ0(t)))| = g(γ0(t))
holds almost everywhere for t ∈ [0, `]. It follows that
|u0(`)− u0(0)| ≤ˆ `
0
|u′0(t)| dt
≤ˆ `
0
g(γ0(t)) dt
=
ˆγ0
g.
(4.1.4)
37
Combining inequalities (4.1.2), (4.1.3), (4.1.4), we get
|u(γ(a))− u(γ(b))| = |u(γ0(0))− u(γ0(`))|
= |u0(0)− u0(`)|
≤ˆγ0
g
≤ˆγ
g.
(4.1.5)
Thus, we verify that g ∈ L1(X) is an upper gradient of u.
If u ∈ AC(X) ∩ L1(X), then Theorem 4.1.3 implies that u ∈ N1,1(X). If we further
assume that u ∈ ACp(X), that is, the upper gradient g defined in (4.1.1) belongs to Lp(X).
It follows that u ∈ N1,p(X) and ACp(X) ⊂ N1,p(X). On the other hand, we need the
following lemma [59, Proposition 6.3.3].
Lemma 4.1.4. Let u : X → R be a function and γ : [0, `]→ X be an arc-length parametrized
rectifiable curve in X. Assume that ρ : X → [0,∞] is a Borel function such that ρ is
integrable on γ and the pair (u, ρ) satisfies the upper gradient inequality on γ and each of its
compact subcurves. Then u γ is absolutely continuous and the inequality
|(u γ)′(t)| ≤ (ρ γ)(t),
holds for almost every t ∈ [0, `]
If u ∈ N1,p(X), then it has a p-integrable upper gradient ρ and (u, ρ) satisfies the upper
gradient inequality for any arc-length parametrized simple rectifiable curves in X, that is,
|u(γ(`))− u(γ(0))| ≤ˆ `
0
ρ(γ(s))ds.
The absolute continuity of integral implies immediately that u ∈ AC(X).
Moreover, if u ∈ N1,p(X), the upper gradient g defined in (4.1.1) is p-integrable. Let
u ∈ N1,p and γi be the simple curves as in the proof of Theorem 4.1.3. Lemma 4.1.4 implies
that
|(u γi)′(t)| ≤ (ρ γi)(t).
38
For almost everywhere x ∈ X, the above inequality implies that
g(x) = |(u γi)′(γ−1i (x))| ≤ ρ(x).
The upper gradient g defined in (4.1.1) is bounded by a p-integrable function ρ almost
everywhere in X. It implies that g ∈ Lp(X) and u ∈ ACp(X). Combining the above facts,
we get the following characterization.
Theorem 4.1.5. Let (X, d,H1) be a compact and connected metric measure space with
H1(X) <∞ and let 1 ≤ p <∞. Then u ∈ N1,p(X) if and only if u ∈ ACp(X) ∩ Lp(X). In
other words
N1,p(X) = ACp(X) ∩ Lp(X).
4.1.3 Spaces supporting the Poincare inequality
In this section, we will show that a compact, quasiconvex and 1-Ahlfors regular metric
measure space supports Poincare inequality. Thus, it implies the equivalence of absolutely
continuous functions with Sobolev functions defined via different approaches.
Theorem 4.1.6. Let (X, d,H1) be a compact, quasiconvex and 1-Ahlfors regular metric
measure space. Then it supports p-Poincare inequality for 1 ≤ p <∞.
Proof. It suffices to show that X supports the 1-Poincare inequality. Let B(O, r) ⊂ X be an
arbitrary ball in X and x, y ∈ B. There exists a rectifiable curve that joins x and y. Since X
is compact and quasiconvex, there exists a shortest curve connecting x and y [47, Theorem
3.9]. We denote the arc-length parametrization of this shortest curve as γ : [0, `]→ X such
that γ(0) = x and γ(`) = y.
Let u be a Borel function and g be an upper gradient of u. Then
|u(x)− u(y)| ≤ˆ `
0
g(γ(s)) ds. (4.1.6)
Denote Γ = γ([0, `]) and λ = 3C, where C is the quasiconvex constant of X.
39
For an arbitrary point z ∈ Γ, let `γ(x,z) denotes the length of the shortest curve joining
x and z. Then `γ(x,z) ≤ `. Since X is quasiconvex, there is a curve γ(x, y) connecting x and
y such that `γ(x,y) ≤ Cd(x, y). It implies that
d(z,O) ≤ d(z, x) + d(x,O)
≤ `γ(x,z) + r
≤ `+ r
≤ `γ(x,y) + r
≤ Cd(x, y) + r
≤ λr.
Thus, every point z ∈ Γ belongs to the ball λB = B(O, λr). It implies that
ˆ `
0
g(γ(s)) ds =
ˆΓ
g(y) dH1(y)
≤ˆλB
g(y) dH1(y).
(4.1.7)
Finally, combining (4.1.6), (4.1.7), we get
|u(x)− uB| ≤ B
|u(x)− u(y)| dz
≤ B
ˆ `
0
g(γ(s)) dsdz
≤ B
ˆλB
g(y) dH1(y)dz
= H1(λB)
λB
g(y) dH1(y).
(4.1.8)
Since X is 1-Ahlfors regular, it follows that
B
|u(x)− uB| dH1 ≤ Cdiam(B)
λB
g(z) dH1(z).
40
Remark 4.1.7. In fact, instead of assuming quasiconvexity, in the proof of this theorem, it
suffice to assume a weaker condition that the space X is C-LLC1, that is, there exists a
constant C ≥ 1 so that for each x ∈ X and r > 0, any pair of points in B(x, r) can be
joined by a curve in B(x,Cr). With this condition, we still have that the shortest curve
connecting any two points x and y is contained in B(x,Cd(x, y)) and the argument in the
above theorem works.
When a complete doubling metric measure X supports the p-Poincare inequality with
1 < p < ∞, the Newton Sobolev spaces N1,p(X), the Poincare Sobolev spaces P 1,p(X) and
Haj lasz Sobolev spaces M1,p(X) are equivalent [47, Theorem 11.3]. Combined with Theorem
4.1.5, we obtain the following corollary.
Corollary 4.1.8. Let (X, d,H1) be a compact, quasiconvex, 1-Ahlfors regular metric measure
space. Then
ACp(X) ∩ Lp(X) = N1,p(X) = P 1,p(X) = M1,p(X),
for 1 < p <∞ and the first two equalities also hold for p = 1.
4.2 ON S-AHLFORS REGULAR METRIC SPACES WITH S ≤ 1
The theory of Newton Sobolev spaces and Poincare Sobolev spaces apply very well to the
spaces with sufficiently many nonconstant rectifiable curves. However, when comes to the
spaces with limited connectivity properties, the theory of Haj lasz Sobolev spaces is rich
without any assumption on connectivity of the space. Haj lasz Sobolev spaces on fractals
have been investigated in [61, 102, 106, 120].
In this section, we will investigate the Haj lasz Sobolev functions in the critical case
M1,s on s-Ahlfors regular spaces with 0 < s ≤ 1. An important class of Ahlfors regular
spaces is provided by fractals. Many fractals are generated by iterated function systems. A
contraction S on D ⊂ Rn is a Lipschitz mapping with Lipschitz constant 0 < c < 1, i.e.
|S(x)− S(y)| ≤ c|x− y|,
41
for all x, y ∈ D. A finite family of contractions S1, · · · , Sm, with m ≥ 2, is called an
iterated function system or IFS. We say that Si satisfy the open set condition if there exists
a non-empty bounded open set V such that
m⋃i=1
Si(V ) ⊂ V
with the union disjoint. Hutchinson [64] proved that fractals generated by iterated function
systems satisfying the open set condition are Ahlfors regular. For example, the ternary
Cantor set C equipped with the Euclidean distance is s-Ahlfors regular with s = log 2log 3
[35,
Example 2.7].
The main theorem of this section states as follows.
Theorem 4.2.1. Let (X, d,Hs) be an s-Ahlfors regular metric space and 0 < s ≤ 1. If
u ∈M1,s(X, d,Hs), then u is uniformly continuous. Moreover, there exists a constant C > 0,
such that for any ball B ⊂ X,
supx,y∈B
|u(x)− u(y)| ≤ C(ˆ
2B
gsdHs) 1
s, (4.2.1)
where g ∈ D(u) ∩ Ls(X).
The restriction that s ≤ 1 is necessary in our statement, since we apply the reverse
Minkowski inequality in the proof. We conjecture that there always exist discontinuous
Sobolev functions in M1,s(X) when X is an s-Ahlfors regular space with s > 1.
We need the following lemma.
Lemma 4.2.2. Let X be a uniformly perfect space. Then there is a constant 0 < C0 < 1
such that for any ball B(x, r) with 0 < r < diam(X), there is a sequence of balls Bi∞i=1 =
B(xi, Ci0r)∞i=1 ⊂ B(x, r) satisfying
(1) Bi ∩Bj = ∅, if i 6= j,
(2) Bi ⊂ B(x,Ci−10 r) \B(x,Ci
0r).
42
Proof. Since X is uniformly perfect, there exists a constant 0 < c < 1 such that for all x ∈ X
and 0 < r < diam(X),
B(x, r) \B(x, cr) 6= ∅.
Fix a number C1 = c/2. Clearly, 0 < C1 < 1.
Fix a ball B(x, r) ⊂ X with 0 < r < diam(X) and let y ∈ B(x, r) \B(x, cr). Then
B(y, C1r) ⊂ B(x,C−11 r) \B(x,C1r).
We can construct a sequence of balls B1, B2 . . . ⊂ B(x, r) in the following way. Let
ri = C2i−11 r and Bi = B(x, ri). By the previous argument, for each Bi, there exists xi such
that
B(xi, C1ri) ⊂ B(x,C−11 ri) \B(x,C1ri).
Let ri = C1ri = C2i1 r, so
B(xi, ri) ⊂ B(x, ri−1) \B(x, ri).
Let C0 = C21 . Clearly, 0 < C0 < 1. Let Bi = B(xi, ri) = B(xi, C
i0r). Since these balls
are contained in disjoint annuli, Bi ∩Bj = ∅ for i 6= j.
From the construction, the balls in this sequence satisfy
Bi ⊂ B(x,Ci−10 r) \B(x,Ci
0r).
This completes the proof.
We now can complete the proof of the main result, following some ideas from [45].
Proof of Theorem 4.2.1. By definition, for u ∈ M1,s(X, d,Hs), there exists g ∈ Ls(X) such
that
|u(x)− u(y)| ≤ d(x, y)(g(x) + g(y)
),
when x, y ∈ X \ E and Hs(E) = 0.
Let E0 = x ∈ X \ E : g(x) < ∞. Clearly, Hs(X \ E0) = 0. Fix a ball B(z, r) ⊂ X.
We will prove that
supx,y∈B(z,r)∩E0
|u(x)− u(y)| ≤ C( ˆ
B(z,2r)
gsdHs) 1
s.
43
Let x, y ∈ B(z, r) ∩ E0. According to Lemma 4.2.2, there exists a sequence of disjoint
balls Bi∞i=1 = B(xi, Ci0r)∞i=1 ⊂ B(x, r) such that
Bi = B(xi, Ci0r) ⊂ B(x,Ci−1
0 r) \B(x,Ci0r).
Let ri = Ci0r, then
Bi = B(xi, ri) ⊂ B(x,C−10 ri) \B(x, ri).
Clearly, for each i ∈ N, we can find zi ∈ Bi ∩ E0 such that
g(zi) ≤( 1
Hs(Bi)
)1/s(ˆBi
gsdHs)1/s
≤ C
ri
(ˆBi
gsdHs)1/s
.
Notice that zi ∈ Bi ⊂ B(x,C−10 ri) implies that d(x, zi) ≤ C−1
0 ri. It follows that
|u(x)− u(zi)| ≤ d(x, zi)(g(x) + g(zi)
)≤ Crig(x) + C
(ˆBi
gsdHs)1/s
.
Hence, limi→∞ u(zi) = u(x). Thus,
|u(z1)− u(x)| ≤∞∑i=1
|u(zi)− u(zi+1)|
=∞∑i=0
d(zi, zi+1)(g(zi) + g(zi+1)
). (4.2.2)
Since zi ∈ Bi, zi+1 ∈ Bi+1 and Bi, Bi+1 ⊂ B(x,C−10 ri), we have that
d(zi, zi+1) ≤ Cri.
Moreover, ri+1 = C0ri and so the inequality (4.2.2) implies that
|u(z1)− u(x)| ≤∞∑i=0
Cri
( 1
ri
( ˆBi
gsdHs)1/s
+1
ri+1
( ˆBi+1
gsdHs)1/s)
≤ C∞∑i=0
(ˆBi
gsdHs)1/s
≤ C( ∞∑i=0
ˆBi
gsdHs)1/s
≤ C( ˆ
B(x,r)
gsdHs)1/s
. (4.2.3)
44
The second to last inequality follows from the reverse Minkowski inequality and the last
inequality follows from the fact that the balls in the sequence Bi∞i=1 ⊂ B(x, r) are pairwise
disjoint.
Notice that z1 ∈ B1 ⊂ B(x, r) ⊂ B(z, 2r) and
g(z1) ≤ C
r1
(ˆB1
gsdHs)1/s
≤ C
r
(ˆB(z,2r)
gsdHs)1/s
.
Similarly, we can find w1 ∈ B(y, r) ⊂ B(z, 2r) such that
|u(w1)− u(y)| ≤ C(ˆ
B(y,r)
gsdHs)1/s
, (4.2.4)
and
g(w1) ≤ C
r
( ˆB(z,2r)
gsdHs)1/s
.
Since z1, w1 ∈ B(z, 2r), we also have
|u(z1)− u(w1)| ≤ d(z1, w1)(g(z1) + g(w1))
≤ C( ˆ
B(z,2r)
gsdHs)1/s
. (4.2.5)
Applying triangle inequality and combining the inequalities (4.2.3), (4.2.4), (4.2.5), we
have
supx,y∈B(z,r)∩E0
|u(x)− u(y)| ≤ C( ˆ
B(z,2r)
gsdHs) 1
s.
This and the absolute continuity of the integral imply uniform continuity of u|X\E0 . Hence
u uniquely extends to a uniformly continuous function on X and the inequality (4.2.1)
follows.
Remark 4.2.3. A closer inspection of the proof of the main theorem suggests that the estimate
(4.2.1) also holds under the weaker conditions that the metric space (X, d,Hs) is uniformly
perfect and satisfies the condition that there exists a constant C > 0 such that for all x ∈ X
and 0 < r < diam(X),
Hs(B(x, r)) ≥ Crs.
However, to get the uniform continuity of the function by this inequality (4.2.1), the
condition that Hs(B(x, r)) ≤ Crs seems necessary.
45
5.0 SOBOLEV EMBEDDING OF A SPHERE CONTAINING AN
ARBITRARY CANTOR SET IN THE IMAGE
Let Sn denotes the standard unit sphere in Rn+1. In 1924 J. W. Alexander [1], constructed a
homeomorphism f : S2 → f(S2) ⊂ R3 so that the unbounded component of R3 \ f(S2) is not
simply connected. In particular it is not homeomorphic to the complement of the standard
ball in R3. This famous construction known as the Alexander horned sphere can be easily
generalized to higher dimensions. The aim of this chapter is to show that a large class of
pathological topological n-dimensional spheres, including the Alexander horned sphere, can
be realized as the images of Sobolev W 1,n homeomorphisms, each of which being a smooth
diffeomorphism outside of a Cantor set in Sn of Hausdorff dimension zero.
5.1 INTRODUCTION
Recall also that the Sobolev space W 1,p consists of functions in Lp whose distributional
gradient is in Lp. By f ∈ W 1,p(Sn,Rn+1) we will mean that the components of the mapping
f : Sn → Rn+1 are in W 1,p.
By an embedding we will mean a homeomorphism onto the image i.e., f : X → Y is an
embedding if f : X → f(X) is a homeomorphism. In the literature such an embedding is
often called a topological embedding. The main result of this chapter reads as follows.
Theorem 5.1.1. For any Cantor set C ⊂ Rn+1, n ≥ 2, there is an embedding f : Sn → Rn+1
such that
(a) f ∈ W 1,n(Sn,Rn+1),
46
(b) C ⊂ f(Sn),
(c) f−1(C) ⊂ Sn is a Cantor set of Hausdorff dimension zero,
(d) f is a smooth diffeomorphism in Sn \ f−1(C).
Our construction resembles that of the Alexander horned sphere and it will be clear that
it can be used to construct a version of the Alexander horned sphere f : Sn → f(Sn) ⊂ Rn+1
so that f ∈ W 1,n and f is a smooth diffeomorphism outside a Cantor set of Hausdorff
dimension zero.
A similar technique to the one used in the proof of Theorem 5.1.1 has also been employed
in a variety of different settings [15, 14, 22, 44, 51, 50, 66, 67, 71, 94, 104, 103, 116, 118, 117].
According to Theorem 5.1.1 we can construct a topological sphere in R3 that is W 1,2
homeomorphic to S2 and that contains Antoine’s necklace. Antoine’s necklace is a Cantor
set in R3 whose complement is not simply connected. Hence the unbounded component
of R3 \ f(S2) is also not simply connected. This gives a different example with the same
topological consequences as those of the Alexander horned sphere. In fact, using results of
Sher [113] we will show that there are uncountably many “essentially different” examples.
For a precise statement see Theorem 5.5.1.
One cannot in general demand the function constructed in the theorem to be in W 1,p, p >
n. Indeed, if f ∈ W 1,p(Sn,Rn+1), then the image f(Sn) has finite n-dimensional Hausdorff
measure, but a Cantor set in Rn+1 may have positive (n + 1)-dimensional measure and in
that case it cannot be contained in the image of f . The fact that the image f(Sn) has finite
n-dimensional measure follows from the area formula and the integrability of the Jacobian
of f . The fact that the area formula is satisfied for mappings f ∈ W 1,p(Sn,Rn+1) with
p > n is well known and follows from the following observations. The area formula is true
for Lipschitz mappings [33, Theorem 3.3.2]. The domain Sn is the union of countably many
sets on which f is Lipschitz continuous [33, Section 6.6.3] plus a set of measure zero. On
Lipschitz pieces the area formula is satisfied. Since the mapping f maps sets of measure
zero to sets of n-dimensional Hausdorff measure zero [60, Theorem 4.2], the area formula is
in fact true for f . The proof presented in [60, Theorem 4.2] is in the case of mappings into
Rn, but the same proof works in the case of mappings into Rn+1.
It is well known, [60, Theorem 4.9], that any homeomorphism f : Ω → f(Ω) ⊂ Rn of
47
class W 1,n, where Ω ⊂ Rn is open, has the Lusin property, i.e. it maps sets of measure
zero to sets of measure zero. Reshetnyak [105] observed that this is no longer true for
embeddings f ∈ W 1,n(Sn,Rm), when m > n ≥ 2. In his example he considered n = 2
and m = 3, see also [28, Example 5.1]. Later Vaisala [115] generalized it to any n ≥ 2
and m > n. In the constructions of Reshetnyak and Vaisala a set of measure zero is
mapped to a set of positive n-dimensional Hausdorff measure. Theorem 5.1.1 also provides
an example of this type. Indeed, if a Cantor set C ⊂ Rn+1 has positive (n+ 1)-dimensional
measure, then the embedding f ∈ W 1,n(Sn,Rn+1) from Theorem 5.1.1 maps the set f−1(C)
of Hausdorff dimension zero onto the set C ⊂ f(Sn) of positive (n+ 1)-dimensional measure.
Actually, in a context of the Lebesgue area a similar example has already been constructed by
Besicovitch [15, 14], but Besicovitch did not consider the Sobolev regularity of the mapping.
The construction of Besicovitch is very different from that of Reshetnyak and Vaisala and
it is more related to ours. While Besicovitch’s construction deals with a particular Cantor
set, we deal with any Cantor set and we prove that the resulting mapping f belongs to the
Sobolev space W 1,n.
5.2 CANTOR SETS AND TREES
The following classical result [119, Theorem 30.3] provides a characterization of spaces that
are homeomorphic to the ternary Cantor set: A metric space is homeomorphic to the ternary
Cantor set if and only if it is compact, totally disconnected and has no isolated points. Recall
that the space is totally disconnected if the only non-empty connected subsets are one-point
sets. In what follows by a Cantor set we will mean any subset of Euclidean space that is
homeomorphic to the ternary Cantor set.
5.2.1 Ternary Cantor set
The ternary Cantor set will be denoted by C. It is constructed by removing the middle third
of the unit interval [0, 1], and then successively deleting the middle third of each resulting
48
subinterval. Denote by Ik all binary numbers i1 . . . ik such that ij ∈ 0, 1 for j = 1, 2, . . . , k,
and by I∞ all binary infinite sequences i1i2 . . . Clearly, the ternary Cantor set C can be
written as
C =∞⋂k=1
⋃i1...ik∈Ik
Ii1...ik ,
where Ii1...ik is one of the 2k closed intervals in the k-th level of the construction of the Cantor
set C; the binary number i1 . . . ik denotes the position of this interval: If ik = 0, it is the left
subinterval of Ii1...ik−1, otherwise it is the right subinterval.
We also have that for any k ∈ N
C =⋃
i1...ik∈IkCi1...ik
where Ci1...ik = C ∩ Ii1...ik .
Note that points in the Cantor set C can be uniquely encoded by infinite binary sequences.
Indeed, if i = i1i2 . . . ∈ I∞, then
ci =∞⋂k=1
Ii1...ik =∞⋂k=1
Ci1...ik
consists of a single point ci ∈ C and
C =⋃i∈I∞ci.
5.2.2 Cantor trees
Let C ⊂ Rn+1 be a Cantor set and let f : C → C be a homeomorphism. We will write
Ci1...ik = f(Ci1...ik) and ci = f(ci) for i ∈ I∞. Since the mapping f is uniformly continuous,
maxi1...ik∈Ik
(diamCi1...ik)→ 0 as k →∞. (5.2.1)
For each k and each i1 . . . ik ∈ Ik we select a point Ai1...ik such that
• The point Ai1...ik does not belong to C,
• The distance of the point Ai1...ik to Ci1...ik is less than 2−k,
• Ai1...ik 6= Aj1...j` if i1 . . . ik 6= j1 . . . j`.
49
It is easy to see that if i = i1i2 . . . ∈ I∞, then
Ai1...ik → ci = f(ci) as k →∞.
Indeed, ci ∈ Ci1...ik so
|Ai1...ik − ci| < 2−k + diamCi1...ik → 0 as k →∞.
Now we are ready to build a Cantor tree by adding branches Ji1...ikik+1connecting Ai1...ik to
Ai1...ikik+1. The precise construction goes as follows.
By translating the coordinate system we may assume that the distance between the
origin in Rn+1 and the Cantor set C is greater than 100 (that is way too much, but there is
nothing wrong with being generous).
Let J0 and J1 be smooth Jordan arcs (smoothly embedded arcs without self-intersections)
of unit speed (i.e., parametrized by arc-length) connecting the origin 0 to the points A0 and
A1 respectively. We also assume that
• The curve J0 does not intersect with the curve J1 (except for the common endpoint 0).
• The curves J0 and J1 avoid the Cantor set C.
• The curves J0 and J1 meet the unit ball Bn(0, 1) ⊂ Rn × 0 ⊂ Rn+1 lying in the
hyperplane of the first n coordinates only at the origin and both curves exit Bn(0, 1) on
the same side of Bn(0, 1) in Rn+1.
The curves J0 and J1 will be called branches of order 1.
A simple topological observation is needed here. While the topological structure of a
Cantor set C inside Rn+1 may be very complicated (think of Antoine’s necklace), no Cantor
set can separate open sets in Rn+1. Indeed, by [63, Corollary 2 of Theorem IV 3] compact sets
separating open sets in Rn+1 must have topological dimension at least n but the topological
dimension of a Cantor set is 0, [63, Example II 3]. Hence we can connect points in the
complement of a Cantor set by smooth Jordan arcs that avoid the Cantor set. Moreover we
can construct such an arc in a way that it is arbitrarily close to the line segment connecting
the endpoints.
50
Recall that the ε-neighborhood of a set A is the set of all points whose distance to the
set A is less than ε.
Suppose that we have already constructed all branches Ji1...ik of order k ≥ 1. The
construction of the branches of order k + 1 goes as follows. Ji1...ik+1 is a family of 2k+1
curves such that
• Ji1...ik+1is a smooth Jordan arc parametrized by arc-length that connects Ai1...ik (an
endpoint of the branch Ji1...ik) to Ai1...ikik+1.
• The curves Ji1...ik+1do not intersect with the Cantor set C, they do not intersect with
each other (except for the common endpoints) and they do not intersect with previously
constructed branches of orders less than or equal to k (except for the common endpoints).
• The image of the curve Ji1...ik+1is contained in the 2−k-neighborhood of the line segment
Ai1...ikAi1...ik+1.
• The angle between the branch Ji1...ik and each of the emerging branches Ji1...ik0 and
Ji1...ik1 at the point Ai1...ik where the curves meet is larger than π/2.
The reason why we require the last condition about the angles is far from being clear at the
moment, but it will be clarified in the end of this section.
In what follows, depending on the situation, Ji1...ik will denote either the curve (a map
from an interval to Rn+1) or its image (a subset of Rn+1), but it will always be clear from
the context what interpretation we use.
A Cantor tree is the closure of the union of all branches
T =∞⋃k=1
⋃i1...ik∈Ik
Ji1...ik .
We also define Tk to be the tree with branches of orders less than or equal to k removed.
Formally
Tk = T \Bk where Bk =k⋃s=1
⋃i1...is∈Is
Ji1...is .
Note that the set Tk is not closed – it does not contain the endpoints Ai1...ik .
The branches Ji1...ik+1are very close to the sets Ci1...ik in the following sense.
51
Lemma 5.2.1. A branch Ji1...ik+1is contained in the 2−k+2 + diamCi1...ik neighborhood of
Ci1...ik .
Proof. A branch Ji1...ik+1connects the points Ai1...ik and Ai1...ik+1
. The distance of Ai1...ik and
Ai1...ik+1to the set Ci1...ik is less than 2−k (because Ci1...ik+1
⊂ Ci1...ik). Hence
|Ai1...ik − Ai1...ik+1| < 2 · 2−k + diamCi1...ik
so by the triangle inequality the line segment Ai1...ikAi1...ik+1is contained in the 3 · 2−k +
diamCi1...ik neighborhood of Ci1...ik . Since Ji1...ik+1is contained in the 2−k neighborhood of
the line segment, the lemma follows.
Corollary 5.2.2. Tk is contained in the
εk := 2−k+2 + maxi1...ik∈Ik
(diamCi1...ik)
neighborhood of the Cantor set C and εk → 0 as k →∞.
Proof. Indeed, the branches of Tk are of the form Ji1...is+1 , s ≥ k. Each such branch is
contained in the 2−s+2 + diamCi1...is neighborhood of Ci1...is ⊂ C. Since s ≥ k and Ci1...is ⊂
Ci1...ik we have
2−s+2 + diamCi1...is ≤ 2−k+2 + diamCi1...ik
so Tk is contained in the εk neighborhood of C. The fact that εk → 0 follows from (5.2.1).
The proof is complete.
Since the sets Bk are compact and their complements Tk = T \Bk are in close proximity
of C by Corollary 5.2.2, it easily follows that
T = C ∪∞⋃k=1
⋃i1...il∈Ik
Jii...ik .
The idea of the proof of Theorem 5.1.1 is to build a surface that looks very similar to the
tree T with one dimensional branches Ji1...ik of the tree T replaced by smooth thin surfaces
built around the curves Ji1...ik ; such surfaces will be called tentacles. The parametric surface
f : Sn → Rn+1 will be constructed as a limit of smooth surfaces fk : Sn → Rn+1. This
sequence will be defined by induction. In the step k we replace all branches Ji1...ik by smooth
52
surfaces – tentacles. Such surfaces will be very close to the branches Ji1...ik and they will pass
through the endpoints Ai1...ik . The only place where the set Tk gets close to the branch Ji1...ik
is the endpoint Ai1...ik where the two branches Ji1...ik0 and Ji1...ik1 of the set Tk emerge. The
surface around Ji1...ik , and passing through the point Ai1...ik will be orthogonal to the curve
Ji1...ik at the point Ai1...ik . Since the branches Ji1...ik0 and Ji1...ik1 emerging from that point
form angles larger than π/2 with Ji1...ik the surface will not intersect the branches Ji1...ik0
and Ji1...ik1. By making the surfaces around Ji1...ik thin enough we can make them disjoint
from the set Tk (note that Ai1...ik does not belong to Tk).
5.3 SOBOLEV TENTACLES
It is well known and easy to prove that η(x) = log | log |x|| ∈ W 1,n(Bn(0, e−1)). Define the
truncation of η between levels s and t, 0 < s < t <∞ by
ηts(x) =
t− s if η(x) ≥ t,
η(x)− s if s ≤ η(x) ≤ t
0 if η(x) ≤ s.
Fix an arbitrary τ > 0. For every δ > 0 there is a sufficiently large s such that ηδ,τ := ηs+τs
is a Lipschitz function on Rn with the properties:
supp ηδ,τ ⊂ Bn(0, δ/2),
0 ≤ ηδ,τ ≤ τ and ηδ,τ = τ in a neighborhood B(0, δ′) of 0,ˆRn
|∇ηδ,τ |n < δn.
The function ηδ,τ is not smooth because it is defined as a truncation, however, mollifying
ηδ,τ gives a smooth function, denoted by ηδ,τ , with the same properties as those of ηδ,τ listed
above. In particular ˆRn
|∇ηδ,τ |n < δn. (5.3.1)
53
The graph of ηδ,τ restricted to the ball Bn(0, δ) is contained in the cylinder
(x1, . . . , xn+1)|x21 + . . .+ x2
n ≤ δ2, 0 ≤ xn+1 ≤ τ, (5.3.2)
and it forms a slim “tower” around the xn+1-axis. The function ηδ,τ equals zero in the annulus
Bn(0, δ) \ Bn(0, δ/2) and equals τ in the ball Bn(0, δ′).
Consider now a smooth Jordan arc γ : [−1, τ + 1] → Rn+1 parametrized by arc-length.
We want to construct a smooth mapping γδ : Bn(0, δ) → Rn+1 whose image will be a
smooth, thin, tentacle-shaped surface around the curve γ|[0,τ ]. To do this we will apply a
diffeomorphism Φ mapping the cylinder (5.3.2) onto a neighborhood of the image of the curve
γ. The tentacle-like surface will be the image of the graph of ηδ,τ under the diffeomorphism
Φ.
The construction of a diffeomorphism Φ follows a standard procedure. Let
v1, . . . , vn : [−1, τ + 1]→ TRn+1
be a smooth orthonormal basis in the orthogonal complement of the tangent space to the
curve γ, i.e., for every t ∈ [−1, τ+1], 〈v1(t), . . . , vn(t), γ′(t)〉 is a positively oriented orthonor-
mal basis of Tγ(t)Rn+1. Now we define
Φ(x1, . . . , xn+1) = γ(xn+1) +n∑i=1
xivi(xn+1) for x ∈ Rn+1 with −1 ≤ xn+1 ≤ τ + 1.
Clearly, Φ is smooth and its Jacobian equals 1 along the xn+1 axis, −1 < xn+1 < τ+1. Hence
Φ is a diffeomorphism in a neighborhood of any point on the xn+1-axis, −1 < xn+1 < τ + 1.
Using compactness of the image of the curve γ it easily follows that there is a δ0 > 0 such
that for all 0 < δ < δ0, Φ is a diffeomorphism in an open neighborhood of the cylinder
(5.3.2). Now we define
γδ : Bn(0, δ)→ Rn+1, γδ(x1, . . . , xn) = Φ(x1, . . . , xn, ηδ,τ (x1, . . . , xn)).
Since∂γδ∂xi
=∂Φ
∂xi+
∂Φ
∂xn+1
∂ηδ,τ∂xi
54
it follows that
|Dγδ| ≤√n‖DΦ‖∞(1 + |∇ηδ,τ |),
where ‖DΦ‖∞ is the supremum of the Hilbert-Schmidt norms |DΦ| over the cylinder (5.3.2).
Hence using (5.3.1), for every ε > 0 we can find δ > 0 so small that
ˆBn(0,δ)
|Dγδ|n ≤ C(n)‖DΦ‖n∞δn < ε. (5.3.3)
Observe that δ depends on γ (because ‖DΦ‖∞ depends on γ).
The tentacle γδ maps the annulus Bn(0, δ) \Bn(0, δ/2) onto the isometric annulus in the
hyperplane orthogonal to γ at γ(0). Indeed, for x ∈ Bn(0, δ) \ Bn(0, δ/2), ηδ,τ (x) = 0 and
hence
γδ(x) = Φ(x1, . . . , xn, 0) = γ(0) +n∑i=1
xivi(0)
is an affine isometry. For a similar reason γδ maps the ball Bn(0, δ′) onto the isometric ball
in the hyperplane orthogonal to γ at γ(τ). Finally for x ∈ Bn(0, δ/2) \Bn(0, δ′), γδ creates a
smooth thin surface around the curve γ that connects the annulus at γ(0) with the ball at
γ(τ).
Composition with a translation allows us to define a tentacle γδ : Bn(p, δ) → Rn+1
centered at any point p ∈ Rn.
5.4 MAIN RESULT
We will construct the Sobolev embedded surface f : Sn → Rn+1 as a limit of smooth
embedded surfaces fk : Sn → Rn+1.
By replacing Sn with a diffeomorphic submanifold (still denoted by Sn) we may assume
that it contains the unit ball
Bn = Bn(0, 1) ⊂ Rn × 0 ⊂ Rn+1.
lying in the hyperplane of the first n coordinates.
55
Since the distance of the Cantor set C to the origin is larger than 100, the only parts of
the Cantor tree T that are close to Bn are the branches J0 and J1 that connect the origin to
A0 and A1. Since the branches meet Bn only at the origin and leave Bn on the same side of
Bn, we can assume that Sn meets T only at the origin.
Now we will describe the construction of f1. We want to grow two tentacles from Sn near
the branches J0 and J1 all the way to points A0 and A1, but we want to make sure that the
tentacles do not touch the set T1.
To do this we choose two distinct points p0, p1 ∈ Bn close to the origin and modify the
curves J0 and J1 only near the origin, so that the modified Jordan arcs γ0 and γ1 emerge
from the points p0 and p1 instead of the origin, and they are orthogonal to Bn at the points
p0 and p1. The curves γ0 and γ1 quickly meet with J0 and J1 and from the points where
they meet they coincide with J0 and J1, so all non-intersection properties of the curves are
preserved. Since the curves are modified only at their beginnings, outside that place they
are identical with J0 and J1.
Next, we find δ1 > 0 so small that the balls Bn(p0, δ1) and Bn(p1, δ1) are disjoint and
contained in Bn and that there are disjoint tentacles
γiδ1 : Bn(pi, δ1)→ Rn+1 for i = 0, 1
along the curves γ0 and γ1 such that
ˆB(pi,δ1)
|Dγiδ1|n < 4−n for i = 0, 1.
Observe that γiδ1(pi) = Ai for i = 0, 1.
Note that the images of small balls Bn(p0, δ′1) and Bn(p1, δ
′1) are isometric balls, orthog-
onal to the curves γ0 and γ1 (and hence to the curves J0 and J1) at the endpoints A0 and
A1. Since the branches J00, J01 form angles larger than π/2 with the curve γ0 at A0 and the
branches J10, J11 form angles larger than π/2 with the curve γ1 at A1 we may guarantee, by
making the tentacles sufficiently thin, that they are disjoint from the set T1 (observe that
the endpoints A0 and A1 do not belong to T1).
Also each annulus Bn(pi, δ1) \ Bn(pi, δ1/2) for i = 0, 1 is mapped isometrically by γiδ1
onto an annulus centered at γi(0) = pi in the hyperplane orthogonal to γi at γi(0) = pi.
56
Since the curve γi is orthogonal to Bn at γi(0) = pi, the annulus Bn(pi, δ1) \ Bn(pi, δ1/2) is
mapped isometrically onto itself. By choosing an appropriate orthonormal frame vi1, . . . vin in
the definition of γiδ1 we may assume that γiδ1 is the identity in the annulus. This guarantees
that the mapping
f1(x) =
γ0δ1
(x) if x ∈ Bn(p0, δ1),
γ1δ1
(x) if x ∈ Bn(p1, δ1),
x if x ∈ Sn \ (Bn(p0, δ1) ∪ Bn(p1, δ1))
is continuous and hence smooth. The construction guarantees also that f1 is a smooth
embedding of Sn into Rn+1 with the image that is disjoint from T1.
The mapping f1 maps the small balls Bn(p0, δ′1) and Bn(p1, δ
′1) onto isometric balls cen-
tered at A0 and A1 respectively with f1(pi) = Ai for i = 0, 1. Now the mapping f2 will be
obtained from f1 by adding four more tentacles: from the ball f1(Bn(p0, δ′1)) centered at A0
there will be two tentacles connecting this ball to the points A00 and A01 and from the ball
at f1(Bn(p1, δ′1)) centered at A1 there will be two tentacles connecting this ball to the points
A10 and A11. More precisely the inductive step is described as follows.
Suppose that we have already constructed a mapping fk, k ≥ 1 such that
• fk is a smooth embedding of Sn into Rn+1 whose image is disjoint from Tk.
• There are 2k disjoint balls Bn(pi1...ik , δk) ⊂ Bn and 2k tentacles
γi1...ikδk: Bn(pi1...ik , δk)→ Rn+1
for i1 . . . ik ∈ Ik such that
3 γi1...ikδk(pi1...ik) = Ai1...ik .
3 The image of γi1...ikδkis in the 2−k neighborhood of the curve Ji1...ik .
3 We have ˆBn(pi1...ik ,δk)
|Dγi1...ikδk|n dx < 4−nk. (5.4.1)
• The mapping fk satisfies
fk =
γi1...ikδkin Bn(pi1...ik , δk) for i1 . . . ik ∈ Ik,
fk−1 in Sn \⋃i1...ik∈Ik B
n(pi1...ik , δk).
57
Observe that fk(pi1...ik) = Ai1...ik and that for some small 0 < δ′k < δk, fk maps balls
Bn(pi1...ik , δ′k) onto isometric balls centered at Ai1...ik .
Now we will describe the construction of the mapping fk+1.
For each i1 . . . ik ∈ Ik we choose two points
pi1...ik0, pi1...ik1 ∈ Bn(pi1...ik , δ′k)
and modify the curves Ji1...ik0 and Ji1...ik1 to γi1...ik0 and γi1...ik1 in a pretty similar way as
we did for the curves γ0 and γ1: the new curves γi1...ik0 and γi1...ik1 emerge from the points
fk(pi1...ik0) and fk(pi1...ik1), they are orthogonal to the ball fk(Bn(pi1...ik , δ′k)) at these points
and then they quickly meet and coincide with Ji1...ik0 and Ji1...ik1.
We find δk+1 > 0 so small that
• The balls
Bn(pi1...ik0, δk+1),Bn(pi1...ik1, δk+1) ⊂ Bn(pi1...ik , δ′k)
are disjoint.
• There are tentacles
γi1...ikiδk+1: Bn(pi1...iki, δk+1)→ Rn+1 for i = 0, 1
such that
3 γi1...ikiδk+1(pi1...iki) = Ai1...iki.
3 The image of γi1...ikiδk+1is in the 2−(k+1) neighborhood of the curve Ji1...iki.
3 The tentacles do not intersect and they avoid the set Tk+1
3 We have ˆBn(pi1...iki,δk+1)
|Dγi1...ikiδk+1|n dx < 4−n(k+1).
The condition about the distance of the tentacle to the curve Ji1...iki can be easily guaranteed,
because the curve γi1...iki can be arbitrarily close to Ji1...iki and the tentacle can be arbitrarily
thin.
Because the curves γi1...iki, i = 0, 1, are orthogonal to the balls fk(Bn(pi1...ik , δ′k)) at the
points γi1...iki(0) = fk(pi1...iki), by choosing appropriate orthonormal frames in the definition
of γi1...ikiδk+1we may guarantee one more condition
58
• γi1...ikiδk+1= fk in Bn(pi1...iki, δk+1) \ Bn(pi1...iki, δk+1/2).
We are using here the fact that both γi1...ikiδk+1and fk are isometries in that annulus. Now we
define
fk+1 =
γi1...ikik+1
δkin Bn(pi1...ik+1
, δk+1) for i1 . . . ik+1 ∈ Ik+1,
fk in Sn \⋃i1...ik+1∈Ik+1 Bn(pi1...ik+1
, δk+1).
As before fk+1 is a smooth embedding of Sn into Rn+1 whose image is disjoint from Tk+1.
Let
Wk =⋃
i1...ik∈IkBn(pi1...ik , δk).
Clearly, Wk is a decreasing sequence of compact sets and
E :=∞⋂k=1
Wk
is a Cantor set E ⊂ Bn ⊂ Sn. By making the sequence δk converge to zero sufficiently fast
we may guarantee that the Hausdorff dimension of E equals zero. Similarly as in the case
of the ternary Cantor set, points in the set E can be encoded by infinite binary sequences
For i = i1i1 . . . ∈ I∞ we define
ei =∞⋂k=1
Bn(pi1...ik,δk) so E =⋃i∈I∞ei.
Now we define
f(x) =
x if x ∈ Sn \W1,
fk(x) if x ∈ Wk \Wk+1, k = 1, 2, . . .
ci if x = ei ∈ E, i ∈ I∞.
Recall that ci = f(ci) is a point of the Cantor set C.
Lemma 5.4.1. f = fk when restricted to Sn \Wk+1.
Proof. The lemma can be easily proved by induction. Let f0 = id . By the definition of f ,
f = f0 in Sn \W1. Suppose now that f = fk in Sn \Wk+1. According to the construction
of fk+1, fk+1 = fk in Sn \Wk+1, but the definition of f yields f = fk+1 in Wk+1 \Wk+2 so
f = fk+1 in
(Sn \Wk+1) ∪ (Wk+1 \Wk+2) = Sn \Wk+2.
This proves the lemma.
59
Since each of the mappings fk is a smooth embedding whose image does not intersect
with the Cantor set C it follows from the lemma that f restricted to the open set Sn \ E is
a smooth embedding and f(Sn \E)∩C = ∅. In the remaining Cantor set E, the mapping f
is defined as a bijection that maps E onto C. Therefore the mapping f is one-to-one in Sn
and C ⊂ f(Sn).
It remains to prove that f is continuous and that f ∈ W 1,n.
First we will prove that f ∈ W 1,n. The mapping f is bounded and hence its components
are in Ln. Since the mapping f is smooth outside the Cantor set E of Hausdorff dimension
zero, according to the characterization of the Sobolev space by absolute continuity on lines,
[33, Section 4.9.2], it suffices to show that the classical derivative of f defined outside of E
(and hence a.e. in Sn) belongs to Ln(Sn). We have
ˆSn|Df |n =
ˆSn\E|Df |n =
ˆSn\W1
|Df |n +∞∑k=1
ˆWk\Wk+1
|Dfk|n
≤ˆSn\W1
|Df |n +∞∑k=1
ˆWk
|Dfk|n.
Since f(x) = x in Sn \W1, we do not have to worry about the first term on the right hand
side and it remains to estimate the infinite sum.
Note that fk = γi1...ikδkin Bn(pi1...ik , δk) so (5.4.1) yields
ˆBn
(pi1...ik ,δk)
|Dfk|n =
ˆBn
(pi1...ik ,δk)
|Dγi1...ikδk|n < 4−nk.
Hence ˆWk
|Dfk|n =∑
i1...ik∈Ik
ˆBn
(pi1...ik ,δk)
|Dfk|n < 2k · 4−nk < 2−nk
so∞∑k=1
ˆWk
|Dfk|n <∞∑k=1
2−nk <∞.
This completes the proof that f ∈ W 1,n.
Remark 5.4.2. Replacing the estimate in (5.4.1) by ε4−nk one can easily modify the con-
struction so that the mapping f will have an arbitrarily small Sobolev norm W 1,n.
It remains to prove that f is continuous. We will need
60
Lemma 5.4.3. f(Bn(pi1...ik , δk)) is contained in the
rk := 2−k+4 + diamCi1...ik−1
neighborhood of Ci1...ik−1.
Proof. If x ∈ Bn(pi1...ik , δk) ∩ E, then x = ei for some i ∈ I∞ with the first k binary digits
equal i1, . . . , ik i.e., i = i1 . . . ik . . . Hence
f(x) = f(ei) = ci = ci1...ik... ∈ Ci1...ik−1.
If x ∈ Bn(pi1...ik , δk) \ E, then there is s ≥ k such that
x ∈ Bn(pi1...ikik+1...is , δs) \Ws+1 ⊂ Ws \Ws+1
for some binary numbers ik+1, . . . , is.
Since f = fs in Ws \ Ws+1 and fs = γi1...isδsin Bn(pi1...is , δs) we conclude that f(x) =
γi1...isδs(x).
It remains to show that the image of γi1...isδsis contained in the rk neighborhood of Ci1...ik−1
.
By Lemma 5.2.1, Ji1...is is in the 2−s+3 + diamCi1...is−1 neighborhood of Ci1...is−1 . Also
the image of γi1...isδsis contained in the 2−s neighborhood of Ji1...is so the image of γi1...isδs
is
contained in the
2−s+4 + diamCi1...is−1
neighborhood of Ci1...is−1 . Since Ci1...is−1 ⊂ Ci1...ik−1and
2−s+4 + diamCi1...is−1 ≤ 2−k+4 + diamCi1...ik−1
the lemma follows.
61
Now we are ready to complete the proof of continuity of f . Clearly, f is continuous on
Sn\E so it remains to prove its continuity on the Cantor set E. Let ei ∈ E, i = i1i2 . . . ∈ I∞.
Since f(ei) = ci we need to show that for any ε > 0 there is δ > 0 such that if |x− ei| < δ,
then |f(x)− ci| < ε.
Let ε > 0 be given. Let k be so large that
2−k+4 + 2 diamCi1...ik−1< ε
and let δ > 0 be so small that
Bn(ei, δ) ⊂ Bn(pi1...ik , δk).
If |x−ei| < δ, then x ∈ Bn(pi1...ik , δk) so by Lemma 5.4.3, f(x) belongs to the rk neighborhood
of the set Ci1...ik−1. Since ci ∈ Ci1...ik−1
, the distance |f(x)− ci| is less than
rk + diamCi1...ik−1= 2−k+4 + 2 diamCi1...ik−1
< ε.
The proof is complete. 2
5.5 UNCOUNTABLE INEQUIVALENT EMBEDDINGS
We say that two embeddings f, g : Sn → Rn+1, are equivalent if there is a homeomorphism
h : Rn+1 → Rn+1 such that h(f(Sn)) = g(Sn). In this section, we will prove that there are
uncountably many inequivalent embeddings as in Theorem 5.1.1 in R3.
Theorem 5.5.1. There are uncountably many embeddings f : S2 → R3 of class W 1,2(S2,R3)
which are not equivalent.
62
The theorem can be generalized to higher dimensions, but we consider the case n = 2 only
because our proof is based on a result of Sher [113] about Cantor sets in R3. Generalization of
Theorem 5.5.1 to n ≥ 3 would require a generalization of Sher’s result to higher dimensions.
Since this would be a work of purely technical nature with predicted answer, we do not find
it particularly interesting.
Antoine’s necklace A is a Cantor set that is constructed iteratively as follows: Inside a
solid torus A0 in R3 (iteration 0) we construct a chain A1 (iteration 1) of linked solid tori so
that the chain cannot be contracted to a point inside the torus A0. A1 is a subset of R3, the
union of the linked tori. Iteration n + 1 is obtained from the iteration n by constructing a
chain of tori inside each of the tori of An (i.e. inside each of the connected components of
An). Again An+1 is a subset of R3 – the union of all tori in this step of construction. We
also assume that the maximum of the diameters of tori in iteration n converges to zero as n
approaches to infinity. Antoine’s necklace is the intersection A =⋂∞n=0 An. For more details
we refer to [113].
Antoine’s necklace has the following properties:
• R3 \ A is not simply connected.
• For any x ∈ A and any r > 0, A ∩ B3(x, r) contains Antoine’s necklace.
The first property is well known [99, Chapter 18] while the second one is quite obvious:
B3(x, r) contains one of the tori T of one of the iterations (actually infinitely many of such
tori) and T ∩ A is also Antoine’s necklace because of the iterative nature of the procedure.
We also need the following observation.
Lemma 5.5.2. Let A be Antoine’s necklace and let M be a smooth 2-dimensional surface
in R3. Then A ∩M is contained in the closure of A \M , A ∩M ⊂ A \M .
Proof. Suppose to the contrary that for some x ∈ A∩M , and some r > 0 we have B3(x, r)∩
(A \ M) = ∅. By taking r > 0 sufficiently small we can assume that B3(x, r) ∩ M is
diffeomorphic to a disc. More precisely, there is a diffeomorphism Φ of R3 which maps
B3(x, r) ∩ M onto the ball B2(0, 2) in the xy-coordinate plane. Note that by the second
property listed above B3(x, r)∩A contains Antoine’s necklace denoted by A. Since A∩ (A \
M) = ∅ we have A ⊂ B3(x, r)∩M so Φ(A) ⊂ B2(0, 2) and (B2(0, 2)×R)\Φ(A) is not simply
63
connected. By [99, Theorem 13.7 p.93] there is a homeomorphism h of the ball B2(0, 2) onto
itself in such that Φ(A) is mapped onto the standard ternary Cantor set C on the x-axis. This
homeomorphism can be trivially extended to a homeomorphism of B2(0, 2) × R by letting
H(x, y, z) = (h(x, y), z). Clearly the complement of H(Φ(A)) in B2(0, 2) × R is not simply
connected. On the other hand since H(Φ(A)) = C, the complement of this set is simply
connected in B2(0, 2)× R which is a contradiction.
The key argument in our proof is the following result of Sher [113, Corollary 1] that we
state as a lemma.
Lemma 5.5.3. There is an uncountable family of Antoine’s necklaces Aii∈I such that
for any i, j ∈ I, i 6= j there is no homeomorphism h : R3 → R3 with the property that
h(Ai) = Aj.
For each of the sets Ai, let fi : S2 → R3 be an embedding as in Theorem 5.1.1 with the
property that Ai ⊂ fi(S2) and fi(S2) \Ai is a smooth surface (but not closed). It remains to
prove that for i 6= j there is no homeomorphism h : R3 → R3 such that h(fi(S2)) = fj(S2).
Suppose to the contrary that such a homeomorphism h exists. We will show that h(Ai) = Aj
which is a contradiction with Lemma 5.5.3.
Clearly,
h(Ai) = (h(Ai) ∩ Aj) ∪ (h(Ai) ∩ (fj(S2) \ Aj)).
Since fj(S2) \ Aj is a smooth surface, Lemma 5.5.2 yields
h(Ai) ∩ (fj(S2) \ Aj) ⊂ h(Ai) \ (fj(S2) \ Aj) = h(Ai) ∩ Aj ⊂ Aj
so h(Ai) ∩ (fj(S2) \Aj) = ∅ and hence h(Ai) ⊂ Aj. Applying the same argument to h−1 we
obtain that h−1(Aj) ⊂ Ai so Aj ⊂ h(Ai) and hence h(Ai) = Aj. The proof is complete. 2
64
6.0 SEMILINEAR EVOLUTION EQUATIONS IN THE HEISENBERG
GROUP
Since 1980s, the theory of viscosity solutions for nonlinear partial differential equations has
been developed and applied to a wide range of fields. It provides existence and uniqueness of
weak solutions to a very general class of fully nonlinear equations in the space of continuous
functions [27, 80]. We refer to the book by Koike [80] and the notes by Manfredi [95] for a
brief review of the classical theory of viscosity solutions.
A recent trend is to extend the viscosity solution theory to metric measure spaces. Sub-
Riemannian spaces play a central role in analysis on metric measure spaces and apply to
a wide range of fields including robotic control and image reconstruction. The Heisenberg
group H, which is known as the simplest example of sub-Riemmanian manifolds provides
a good setting for the generalization of the viscosity solution theory. We will give a brief
introduction of Heisenberg group in the first section.
This chapter is concerned with the uniqueness and the Lipschitz and convexity preserving
properties for viscous Hamilton-Jacobi equations on the Heisenberg group H:
ut − tr(A(∇2
Hu)∗) + f(p,∇Hu) = 0 in H× (0,∞), (6.0.1)
u(·, 0) = u0 in H, (6.0.2)
where A is a given 2×2 symmetric positive-semidefinite matrix and the function f : H×R2 →
R satisfies certain assumptions to be made explicit later. Here ∇Hu, (∇2Hu)∗ are respectively
the horizontal gradient and the horizontal symmetrized Hessian of the unknown function u
in space, and u0 is a given locally uniformly continuous function in H.
65
Many of our results in this work also hold for more general fully nonlinear degenerate
parabolic equations of the type
ut + F(p,∇Hu, (∇2
Hu)∗)
= 0 in H× (0,∞) (6.0.3)
under proper regularity assumptions on F . We however focus on (6.0.1) for simplicity of
exposition.
We first show uniqueness of viscosity solutions to the equations with exponential growth
at infinity and then turn to investigate the Lipschitz and convexity preserving of the unique
viscosity solution. It turns out that in general, such properties cannot be expected to hold
in the Heisenberg group. Some restrictions on the class of solutions prove to be necessary.
Thus, we pose several appropriate conditions and prove that the Lipschitz continuity and
horizontal convexity property of the solutions is preserved under these assumptions. On the
other hand, there are also many affirmative examples of Lipschitz and convexity preserving
in the Heisenberg group.
6.1 THE HEISENBERG GROUP H
6.1.1 Definitions and basic properties
The Heisenberg group H is the space R3 endowed with the non-commutative group multi-
plication
(xp, yp, zp) · (xq, yq, zq) =
(xp + xq, yp + yq, zp + zq +
1
2(xpyq − xqyp)
),
for all p = (xp, yp, zp) and q = (xq, yq, zq) in H. Note that the group inverse of p = (xq, yq, zq)
is p−1 = (−xq,−yq,−zq). The Koranyi gauge is given by
|p|G = ((p21 + p2
2)2 + 16p23)1/4,
and the left-invariant Koranyi or gauge metric is
dL(p, q) = |p−1 · q|G.
66
The Lie Algebra of H is generated by the left-invariant vector fields
X1 =∂
∂x− y
2
∂
∂z;
X2 =∂
∂y+x
2
∂
∂z;
X3 =∂
∂z.
One may easily verify the commuting relation X3 = [X1, X2] = X1X2 −X2X1.
The horizontal gradient of u is given by
∇Hu = (X1u,X2u)
and the symmetrized second horizontal Hessian (∇2Hu)∗ ∈ S2×2 is given by
(∇2Hu)∗ :=
X21u (X1X2u+X2X1u)/2
(X1X2u+X2X1u)/2 X22u
.
Here Sn×n denotes the set of all n× n symmetric matrices.
We denote by H0 the set of horizontal vectors of the form (h1, h2, 0), or equivalently
left-invariant vector fields of the form h1X1 + h2X2. The horizontal subspace at a point p
is the two dimensional space spanned by X1(p) and X2(p) and it can be identified with the
left translation of H0 by p, that is,
p ·H0 = Linear− spanX1(p), X2(p).
A piecewise smooth curve s 7→ γ(s) ∈ H is called horizontal if its tangent vector γ′(s) is
in the linear span of X1(γ(s)), X2(γ(s)) for every s such that γ′(s) exists; in other words,
there exist a(s), b(s) ∈ R satisfying
γ′(s) = a(s)X1(γ(s)) + b(s)X2(γ(s))
whenever γ′(s) exists. We denote
‖γ′(s)‖ =(a2(s) + b2(s)
) 12 .
67
Given p, q ∈ H, denote
Γ(p, q) = horizontal curves γ(s) (s ∈ [0, 1]): γ(0) = p and γ(1) = q.
Chow’s theorem states that Γ(p, q) 6= ∅; see, for example, [13]. The Carnot-Caratheodory
metric is then defined to be
dCC(p, q) = infγ∈Γ(p,q)
ˆ 1
0
‖γ′(s)‖ ds.
We next give the Taylor expansion of a smooth function u : H → R. By the fact that
|p|2G ≈ x2 + y2 + |z|, we obtain the horizontal Taylor expansion at point p0 = (x0, y0, z0):
u(p) = u(p0) + 〈∇u(p0), p−10 · p〉+
1
2〈(∇2
Hu(p0))∗h, h〉+ o(|p−10 · p|2), (6.1.1)
where ∇u(p0) is the full gradient of u at p0 and the vector h is a horizontal projection of the
vector p−10 · p.
6.1.2 Metrics on H
Besides the left-invariant Koranyi metric dL and Carnot-Caratheodory metric dCC , the func-
tion dR(p, q) = |p·q−1|G for any p, q ∈ H defines another metric on H, which is right invariant;
in fact, dR(p, q) = dL(p−1, q−1) for any p, q ∈ H.
It is known that dL is bi-Lipschitz equivalent to the Carnot-Caratheodory metric dCC
[21, 95]. The metrics dL and dR are not bi-Lipschitz equivalent, which is indicated in the
example below.
Example 6.1.1. One may choose
p = (1− ε, 1 + ε, ε), q = (1, 1, 0)
with ε > 0 small, then by direct calculation, we have dL(p, q)4 = |p−1 · q|4G = 4ε4 and
dR(p, q)4 = |p · q−1|4G = 4ε4 + 64ε2, which indicates that one cannot expect the existence of
a constant C > 0 such that dR(p, q) ≤ CdL(p, q) for all p, q ∈ H. A variant of this example
shows that the reverse inequality also fails in general.
68
Although the metrics above are not bi-Lipschitz equivalent, it turns out that one is locally
Holder continuous in relative to the other.
Proposition 6.1.2. For any ρ > 0, there exists Cρ > 0 such that
dL(p, q) ≤ CρdR(p, q)12 (6.1.2)
and
dR(p, q) ≤ CρdL(p, q)12 (6.1.3)
for any p, q ∈ H with |p|, |q| ≤ ρ.
Proof. We only show (6.1.2). The proof of (6.1.3) is similar. Set p = (xp, yp, zp) and q =
(xq, yq, zq). It is then clear that we only need to show that there exists some C > 0 depending
only on ρ such that∣∣∣∣zp − zq +1
2xpyq −
1
2xqyp
∣∣∣∣≤ C
((|xp − xq|2 + |yp − yq|2)2 + 16
(zp − zq −
1
2xpyq +
1
2xqyp
)2) 1
4
for all p, q ∈ H with |p|, |q| ≤ ρ.
Let δ = ((|xp − xq|2 + |yp − yq|2)2 + 16|zp − zq − 12xpyq + 1
2xqyp|2)
14 ≤ 1. Then it is clear
that ∣∣∣∣zp − zq +1
2xpyq −
1
2xqyp
∣∣∣∣ ≤ δ2
4+ |xpyq − xqyp|
=δ2
4+ |(xp − xq)yq − xq(yp − yq)| .
It follows that∣∣∣∣zp − zq +1
2xpyq −
1
2xqyp
∣∣∣∣ ≤ δ2
4+ (|xp − xq|2 + |yp − yq|2)
12 (x2
q + y2q )
12 .
Noticing that x2q + y2
q ≤ ρ2, we then have∣∣∣∣zp − zq +1
2xpyq −
1
2xqyp
∣∣∣∣ ≤ δ2
4+ ρ(|xp − xq|2 + |yp − yq|2)
12 ≤ δ2
4+ ρδ =
(δ
4+ ρ
)δ.
We conclude the proof by choosing C = 1/4 + ρ.
69
6.1.3 Lipschitz continuity
We discuss Lipschitz continuity with respect to dL and dR.
It is easily seen that the function f0 : H→ R given by f0(p) = |p|G is a Lipschitz function
with respect to dL and dR, due to the triangle inequality. But there exist functions that are
Lipschitz with respect to one of the metrics but not with respect to the other. An example,
following Example 6.1.1, is as below.
Example 6.1.3. Fix q = (1, 1, 0) ∈ H as in Example 6.1.1. Let fq : H → R defined by
fq(p) = dR(p, q) for every p ∈ H, which satisfies
|fq(p)− fq(p′)| = |dR(p, q)− dR(p′, q)| ≤ dR(p, p′)
for all p, p′ ∈ H. But there is no constant L > 0 such that
fq(p)− fq(p′) ≤ LdL(p, p′)
for all p, p′ ∈ H, for otherwise we may take p = (1− ε, 1 + ε, ε) and p′ = q, and get
dR(p, p′) ≤ LdL(p, p′),
which is not true when ε > 0 small, as explained in Example 6.1.1. However, by Proposition
6.1.2, the function fq is still locally 1/2-Holder continuous with respect to dL.
On the other hand, not all functions that are (locally) Lipschitz with respect to dL or
dR are (locally) Lipschitz with respect to the Euclidean metric. The simplest example is the
function f(p) = |p|G for p ∈ H.
We conclude this section by showing the equivalence of Lipschitz continuity with respect
to both metrics for functions with symmetry. We include in our discussions two different
types of evenness.
Definition 6.1.4 (Even functions). We say a function f : H → R is even (or symmetric
about the origin) if f(p) = f(p−1) for all p ∈ H. We say f is vertically even (or symmetric
about the horizontal coordinate plane) if f(p) = f(p) for all p ∈ H, where
p = (x, y,−z) for any p = (x, y, z) ∈ H. (6.1.4)
70
Since |p · q−1|G = |p−1 · q|G = |(p−1)−1 · q−1|G for any p, q ∈ H, the following result is
obvious.
Proposition 6.1.5 (Equivalence of Lipschitz continuities). Let f : H → R be a function
that is either even or vertically even in H. Then f is Lipschitz continuous with respect to dL
if and only if f is Lipschitz continuous with respect to dR.
6.1.4 Horizontal convexity
For any point p ∈ H, we denote the line segment joining p · h−1 and p · h by [p · h−1, p · h],
which is always a line segment contained in the horizontal subspace p ·H0.
Definition 6.1.6 ([91, Definition 4.1]). Let Ω be an open set in H and u : Ω → R be an
upper semicontinuous function. The function u is said to be horizontally convex or h-convex
in Ω, if for every p ∈ H and h ∈ H0 such that [p · h−1, p · h] ⊂ Ω, we have
u(p · h−1) + u(p · h) ≥ 2u(p). (6.1.5)
One may also define convexity of a function through its second derivatives in the viscosity
sense.
Definition 6.1.7. Let Ω be an open set in H and u : Ω → R be an upper semicontinuous
function. The function u is said to be v-convex in Ω if
(∇2Hu)∗(p) ≥ 0 for all p ∈ H (6.1.6)
in the viscosity sense.
It is clear that u ∈ C2(Ω) is v-convex if it satisfies (6.1.6) everywhere in Ω. It is known
that the h-convexity and v-convexity are equivalent [91]. The following example shows that
h-convexity is very different from convexity in the Euclidean sense.
71
Example 6.1.8. Let
f(x, y, z) = x2y2 + 2z2 (6.1.7)
for all (x, y, z) ∈ H. It is not difficult to verify that f is h-convex. Indeed, for any p =
(x, y, z) ∈ H and h = (h1, h2, 0) ∈ H0, we have
f(p · h) + f(p · h−1)
= 2x2y2 + 4z2 + 3x2h22 + 3y2h2
1 + 2h21h
22 + 6xyh1h2
≥ 2f(p) + 3(xh2 + yh1)2 + 2h21h
22 ≥ 2f(p).
The function f is an example of (globally) h-convex functions in H that is not convex in R3.
6.2 REVIEW OF VISCOSITY SOLUTIONS
6.2.1 Viscosity solutions in Rn
In this section, we review the viscosity solution theory for the general second-order fully
nonlinear equations arising in various fields:
F (x, u(x),∇u(x),∇2u(x)) = 0, (6.2.1)
where x = (x1, x2, . . . , xn) ∈ Rn are independent variables, u : Rn → R is the unknown
function, and ∇u = (ux1 , ux2 , . . . , uxn) and ∇2u respectively denote the gradient and Hessian
of u with respect to x. Here F is a continuous function defined on Rn × R × Rn × Sn×n,
where Sn×n stands for the set of all symmetric n× n matrices.
We are particularly interested in the class of equations with the following assumptions:
(1) (Ellipticity) F (x, r, p,X) is decreasing in X, i.e.,
F (x, r, p,X) ≤ F (x, r, p, Y ) for all X, Y ∈ Sn×n such that X ≥ Y .
(2) (Monotonicity) F (x, r, p,X) is increasing in r.
72
Regarding the notation X ≥ Y , we recall that there is a natural partial ordering in Sn×n;
indeed, X ≥ Y means that X − Y is positive semi-definite. We call F is proper if the above
two conditions are satisfied.
The viscosity approach is particularly useful when F is not in the divergence form, i.e.,
the structure of F does not allow integration. We provide the definition of viscosity solutions
of (6.2.1) in a bounded domain Ω ⊂ Rn.
Definition 6.2.1. We say u ∈ C(Ω) is a viscosity subsolution (resp., supersolution) of (6.2.1)
in Ω if for any test function φ ∈ C2(Ω) and x0 ∈ Ω such that u − φ attains a maximum
(resp., minimum) over Ω at x0, we have
F (x0, u(x0),∇φ(x0),∇2φ(x0)) ≤ 0 (resp., F (x0, u(x0),∇φ(x0),∇2φ(x0)) ≥ 0.)
We call u is a viscosity solution if it is both a viscosity subsolution and a viscosity superso-
lution.
Remark 6.2.2. It is possible to give equivalent definitions by replacing the maximum /min-
imum in the definition above with “strict (unique) maximum/minimum” or “local maxi-
mum/minimum”.
Remark 6.2.3. We usually take C1 test functions if F is a first order operator. In fact,
one may choose to use smooth (C∞) functions as tests in Definition 6.2.1 and get another
equivalent definition [38, Proposition 2.2.3].
Remark 6.2.4. It is also possible to define subsolutions for the space of all upper semicon-
tinuous functions on Ω, denoted by USC(Ω). Similarly, the lower semicontinuous function
space (LSC(Ω)) can be used to define supersolutions.
We may think of the test for subsolutions in Definition 6.2.1 in a more geometric way by
considering that φ touches u from above at x0. If u is a viscosity subsolution or supersolu-
tion, we essentially require the derivatives of the test function φ to fulfill the corresponding
inequality. The next result indicates that viscosity solutions are weak solutions which agree
with classical solutions if their regularity is adequate.
Proposition 6.2.5. Assume that F is proper. Then u is a classical solution of (6.2.1) in Ω
if and only if u is a viscosity solution and u ∈ C2(Ω).
73
There are various approaches to the existence theorem of viscosity solutions to a certain
equation. We list several well-known methods.
1. Vanishing viscosity method: We may add an artificial viscosity in the equation to get a
smoother solution and pass to the limit as the viscosity approaches 0.
2. Optimal control/differential game: We set up an optimal control or game problem, whose
value function represents or approximates the viscosity solution.
3. Perron’s method: The supremum of all subsolutions or the infimum of all supersolutions
with boundary data constraint gives a viscosity solution.
The vanishing viscosity method is one of the classical methods to find solutions to fully
nonlinear equations. One may refer to the early work [26] for more details. Perron’s method
is also classical but was first adopted in the theory of viscosity solutions by Ishii [65].
In general, proving the uniqueness is usually harder than proving the existence and
stability in the study of the viscosity solutions. Thus, the comparison principle is the main
issue of the viscosity solutions. In order to state the comparison principle, we introduce the
following equivalent definitions of viscosity solutions.
Definition 6.2.6. A pair (η,X), where η ∈ Rn and X is an n × n symmetric matrix,
belongs to the second order superjet (resp., subjet) of an upper semicontinuous (resp., lower
semicontinuous) function u at point x0 if
u(x0 + h) ≤ u(x0) + 〈η, h〉+1
2〈X · h, h〉+ o(|h|2)
(resp., u(x0 + h) ≥ u(x0) + 〈η, h〉+1
2〈X · h, h〉+ o(|h|2))
as h→ 0. The collection of all of these pairs, is denoted by J2,+u(x0) (resp., J2,−u(x0)).
We also introduce a sort of closure of jets J2,±u(x0) as the collection of (η,X) ∈ Rn×Sn
such that there exists a sequence xk ∈ Ω and a sequence (ηk, Xk) ∈ J2,±u(xk) satisfying
(xk, u(xk), ηk, Xk)→ (x0, u(x0), η,X)
as k →∞.
Proposition 6.2.7. For u : Ω→ R, the following conditions are equivalent:
74
(1) u is a viscosity subsolution (resp., supersolution)of (6.2.1).
(2) For x ∈ Ω and (η,X) ∈ J2,+u(x) (resp., J2,−u(x)), we have F (x, u(x), η,X) ≤ 0 (resp.,
≥ 0).
(3) For x ∈ Ω and (η,X) ∈ J2,+u(x) (resp., J
2,−u(x)), we have F (x, u(x), η,X) ≤ 0 (resp.,
≥ 0).
We next state the maximum principle for semicontinuous functions, which is essential
for the proof of uniqueness of the viscosity solutions.
Theorem 6.2.8 (Crandall-Ishii-Jensen-Lions). Let Ω be a bounded domain in Rn. Let the
function u be upper semicontinuous and v be lower semicontinuous in Ω. Suppose that for
x ∈ ∂Ω we have
lim supy→x
u(y) ≤ lim infy→x
v(y)
where both sides are not ∞ or −∞ simultaneously. If u− v has an interior local maximum
supΩ
(u− v) > 0
then we have: For τ > 0, we can find points pτ , qτ ∈ Ω such that
(1)
limτ→∞
τφ(pτ − qτ ) = 0,
where φ(p) = |p|2,
(2) there exists a point p ∈ Ω such that pτ → p and
supΩ
(u− v) = u(p)− v(p) > 0,
(3) there exist n× n symmetric matrices Xτ , Yτ and vectors ητ so that
(ητ , Xτ ) ∈ J2,+
(u, pτ ),
and
(ητ , Yτ ) ∈ J2,+
(u, qτ ),
(4) Xτ ≤ Yτ .
As a corollary, we give the comparison principle for the viscosity solutions of (6.2.1).
75
Corollary 6.2.9. Let Ω be a bounded domain in Rn and F in (6.2.1) be proper and Lipschitz.
Let the function u be a subsolution of (6.2.1) and v be a supersolution of (6.2.1) in Ω. Suppose
that for x ∈ ∂Ω we have
lim supy→x
u(y) ≤ lim infy→x
v(y)
where both sides are not ∞ or −∞ simultaneously. Then we have u(x) ≤ v(x) for x ∈ Ω.
6.2.2 Viscosity solutions in H
We also define the viscosity solutions in H via semijets.
Definition 6.2.10. A pair (η,X), where η ∈ R3 and X is an 2 × 2 symmetric matrix,
belongs to the second order superjet (resp., subjet) of an upper semicontinuous (resp., lower
semicontinuous) function u at point p0 if
u(p) ≤ u(p0) + 〈η, p−10 · p〉+
1
2〈X · h, h〉+ o(|p−1
0 · p|2)
(resp., u(p0 · h) ≥ u(p0) + 〈η, h〉+1
2〈X · h, h〉+ o(|p−1
0 · p|2)
as |p−10 · p| → 0, where h is the horizontal projection of p−1
0 · p. The collection of all of these
pairs, is denoted by J2,+u(p0) (resp., J2,−u(p0)).
As in the case of Euclidean spaces, we can also define the closure of jets J2,±u(p0) in a
similar way. Consider a continuous function F (p, u, η,X) : H × R × R3 × S(R2) → R. We
will assume that F is proper, that is, F is increasing in u and decreasing in X.
Definition 6.2.11. A lower semicontinuous (resp., upper semicontinuous) function u is a
viscosity supersolution of the equation
F (p, u(p),∇u(p), (∇2u(p))∗) = 0 (6.2.2)
if whenever (η,X) ∈ J2,−u(p0) (resp., (η,X) ∈ J2,+u(p0)) we have
F (p0, u(p0), η,X) ≥ 0 (resp., F (p0, u(p0), η,X) ≤ 0).
Equivalently, if φ touches u from below (resp., above) at p0, is C2 in X1, X2 and C1 in X3,
then we must have
F (p0, u(p0),∇φ(p0), (∇2φ(p0))∗) ≥ 0 (resp., F (p0, u(p0),∇φ(p0), (∇2φ(p0))∗) ≤ 0).
76
By the continuity of F , J2,±u(p0) in the above definition can be replaced by J2,±u(p0).
A viscosity solution is defined as being both a viscosity subsolution and a viscosity superso-
lution.
The following maximum principle is proved by Bieske [16].
Theorem 6.2.12. Let Ω be a bounded domain in H. Let the function u be upper semicon-
tinuous and v be lower semicontinuous in Ω. Suppose that for x ∈ ∂Ω we have
lim supy→x
u(y) ≤ lim infy→x
v(y)
where both sides are not ∞ or −∞ simultaneously. If u− v has an interior local maximum
supΩ
(u− v) > 0
then we have: For τ > 0, we can find points pτ , qτ ∈ Ω such that
(1)
limτ→∞
τφ(pτ · q−1τ ) = 0,
where φ(p) = x4 + y4 + z2,
(2) there exists a point p ∈ Ω such that pτ → p and
supΩ
(u− v) = u(p)− v(p) > 0,
(3) there exist 2× 2 symmetric matrices Xτ , Yτ and vectors ητ ∈ R3 so that
(ητ , Xτ ) ∈ J2,+
(u, pτ ),
and
(ητ , Yτ ) ∈ J2,+
(u, qτ ),
(4) Xτ ≤ Yτ + o(1) as τ →∞. This condition means if ξ ∈ R2, we have
〈Xτξ, ξ〉 − 〈Yτξ, ξ〉 ≤ a(τ)|ξ|2,
where a(τ)→ 0 as τ →∞.
Similarly, we give the comparison principle for the viscosity solutions of (6.2.2) in H.
77
Corollary 6.2.13. Let Ω be a bounded domain in H and F in (6.2.2) be proper and Lipschitz.
Let the function u be a subsolution of (6.2.2) and v be a supersolution of (6.2.2) in Ω. Suppose
that for x ∈ ∂Ω we have
lim supy→x
u(y) ≤ lim infy→x
v(y)
where both sides are not ∞ or −∞ simultaneously. Then we have u(x) ≤ v(x) for x ∈ Ω.
6.3 UNIQUENESS OF UNBOUNDED SOLUTIONS
Uniqueness of viscosity solutions of various nonlinear equations in the Heisenberg group is
studied in [16, 17, 95, 98] etc. But most of these results are either for a bounded domain
or for bounded solutions. It is less understood when the domain and solution are both
unbounded in the Heisenberg group. To the best of our knowledge, the only known result on
uniqueness for time-dependent equations in this case is due to Haller Martin [53], where a
comparison principle is established for a class of nonlinear parabolic equations including the
horizontal Gauss curvature flow of graphs in the Carnot group. The comparison principle in
[53] is for solutions with polynomial growth at infinity while ours is for exponential growth,
but our assumptions on the structure of the equations are stronger.
In this section, motivated by a Euclidean argument in [11], we present a proof of a
comparison principle for (6.0.1) with exponential growth at space infinity. Our result and
proof are different from those of [53].
We need the following Lipschitz continuity of f .
(A1) There exists L1 > 0 such that
|f(p, w1)− f(p, w2)| ≤ L1|w1 − w2| (6.3.1)
for all p ∈ H and w1, w2 ∈ R2.
(A2) There exists L2(ρ) > 0 depending on ρ > 0 such that
|f(p, w)− f(q, w)| ≤ L2(ρ)|p · q−1|G (6.3.2)
for all p, q ∈ H with |p|, |q| ≤ ρ and all w ∈ R2.
78
Here | · |G denotes the Koranyi gauge in H, i.e.,
|p|G =((x2
p + y2p)
2 + 16z2p
) 14
for all p = (xp, yp, zp) ∈ H. Note that (A2) is not the usual local Lipschitz continuity in
H, since the distance between p, q ∈ H defined by dR(p, q) = |p · q−1|G is invariant only
under right translations and therefore not equivalent to the usual gauge metric give by
dL(p, q) = |p−1 · q|G or the Carnot-Caratheodory metric.
Our comparison principle is as below.
Theorem 6.3.1 (Comparison principle for unbounded solutions). Assume that the Lips-
chitz conditions (A1) and (A2) hold. Let u and v be respectively an upper semicontinuous
subsolution and a lower semicontinuous supersolution of (6.0.1). Assume that for any fixed
T > 0, there exist k > 0 and CT > 0 depending on T such that
u(p, t)− v(p, t) ≤ CT ek〈p〉 (6.3.3)
for all (p, t) ∈ H× [0, T ], where
〈p〉 = (1 + x4 + y4 + 16z2)14 for all p = (x, y, z) ∈ H. (6.3.4)
If u(p, 0) ≤ v(p, 0) for all p ∈ H, then u ≤ v in H× [0,∞).
Proof of Theorem 6.3.1. We aim to show that u ≤ v in H × [0, T ) for any fixed T > 0. By
the growth assumption, there exist k > 0 and CT > 0 satisfying condition (6.3.3). Take an
arbitrary constant β > mink, 1 and then α > 0 to be determined later. Set
g(p, t) = eαt+β〈p〉 (6.3.5)
for (p, t) ∈ H× [0,∞). Recall that 〈p〉 is a function of p ∈ H given in (6.3.4). If p = (x, y, z),
we have by direct calculations
∇H〈p〉 =
(x3 − 4yz
(1 + x4 + y4 + 16z2)34
,y3 + 4xz
(1 + x4 + y4 + 16z2)34
), (6.3.6)
79
which implies that there exists µ > 0 such that
|∇Hg(p, t)| ≤ βµg(p, t) (6.3.7)
for all (p, t) ∈ H× [0,∞).
We assume by contradiction that u(p, t)− v(p, t) takes a positive value at some (p, t) ∈
H× (0,∞). Then there exists σ ∈ (0, 1) such that
u(p, t)− v(p, t)− 2σg(p, t)− σ
T − t
attains a positive maximum at (p, t) ∈ H× [0, T ). For all ε > 0 small, consider the function
Φ(p, q, t, s) = u(p, t)− v(q, s)− σΨε(p, q, t, s)−(t− s)2
ε− σ
T − t
with
Ψε(p, q, t, s) = ϕε(p, q) +K(p, q, t, s),
ϕε(p, q) =1
εdR(p, q)4 =
|p · q−1|4
ε, K(p, q, t, s) = g(p, t) + g(q, s).
Then Φ attains a positive maximum at some (pε, qε, tε, sε) ∈ H2 × [0, T )2. In particular,
Φ(pε, qε, tε, sε) ≥ Φ(p, p, t, t),
which implies that
|pε · q−1ε |4
ε+
(tε − sε)2
ε≤u(pε, tε)− v(qε, sε)− σg(pε, tε)− σg(qε, sε)−
σ
T − tε
−(u(p, t)− v(p, t)− 2σg(p, t)− σ
T − t
).
(6.3.8)
Since, due to (6.3.3), the terms u(pε, tε)− v(qε, tε)− σg(pε, tε)− σg(qε, sε) are bounded from
above uniformly in ε, we have
dR(pε, qε)→ 0 and tε − sε → 0 as ε→ 0. (6.3.9)
80
We notice that pε, qε are bounded, since otherwise the right hand side of (6.3.8) will tend
to −∞. Therefore, by taking a subsequence, still indexed by ε, we have pε, qε → p ∈ H and
tε, sε → t ∈ [0, T ). It follows that
lim supε→0
u(pε, tε)− v(qε, sε)− σg(pε, tε)− σg(qε, sε)−σ
T − tε≤ u(p, t)− v(p, t)− 2σg(p, t)− σ
T − t,
which yields
ϕε(pε, qε)→ 0 as ε→ 0.
Also, it is easily seen that t > 0 and therefore tε, sε > 0 thanks to the condition that
u(·, 0) ≤ v(·, 0) in H.
In order to apply the Crandall-Ishii lemma (cf. [27]) in our current case, let us recall the
definition of semijets adapted to the Heisenberg group: for any (p, t) ∈ H× (0,∞) and any
locally bounded upper semicontinuous function u in H× (0,∞),
P 2,+H u(p, t) =
(τ, ζ,X) ∈ R× R3 × S2×2 : u(q, s) ≤u(p, t) + τ(s− t)
+〈ζ, p−1 · q〉+1
2〈Xh, h〉+ o(|p−1 · q|2G)
,
where h denotes the horizontal projection of p−1 · q. Similarly, we may define
P 2,−H u(p, t) =
(τ, ζ,X) ∈ R× R3 × S2×2 : u(q, s) ≥ u(p, t) + τ(s− t)
+〈ζ, p−1 · q〉+1
2〈Xh, h〉+ o(|p−1 · q|2G)
for any locally bounded lower semicontinuous function u. Also, the closure P2,+
H is the set of
triples (τ, ζ,X) ∈ R× R3 × S2×2 that satisfy the following: there exist (pj, tj) ∈ H× [0,∞)
and (τj, ζj, Xj) ∈ P 2,+H (pj, tj) such that
(pj, tj, u(pj, tj), τj, ζj, Xj)→ (p, t, u(p, t), τ, ζ,X) as j →∞.
The closure set P2,−H of P 2,−
H can be similarly defined. We refer to [17] for more details.
81
We now apply the adaptation of the Crandall-Ishii lemma to the Heisenberg group [95, 17]
and get for any λ ∈ (0, 1)
(a1, ζ1, X) ∈ P 2,+
H u(pε, tε) and (a2, ζ2, Y ) ∈ P 2,−H v(qε, sε)
such that
a1 − a2 = ασK(pε, qε, tε, sε) +σ
(T − tε)2, (6.3.10)
〈Xw,w〉 − 〈Y w,w〉 ≤ 〈(σM + λσ2M2)wpε ⊕ wqε , wpε ⊕ wqε〉, (6.3.11)
and the horizontal projections of ζ1, ζ2 ∈ R3 can be written respectively as ξ + η1 and ξ + η2
(in R2) with
ξ = ∇pHϕε(pε, qε) = −∇q
Hϕε(pε, qε),
η1 = βσ∇Hg(pε, tε), η2 = −βσ∇Hg(qε, sε).
Here w = (w1, w2) ∈ R2 is arbitrary, M = (∇2Ψε)∗(pε, qε, tε, sε) is a 6× 6 symmetric matrix,
and
wpε =
(w1, w2,
1
2w2xpε −
1
2w1ypε
)(6.3.12)
and
wqε =
(w1, w2,
1
2w2xpε −
1
2w1yqε
)(6.3.13)
with pε = (xpε , ypε , zpε) and qε = (xqε , yqε , zqε).
It is easily seen that M = M1 +M2, where
M1 = ∇2ϕε(pε, qε)
and
M2 = ∇2K(pε, qε) =
∇2g(pε, tε) 0
0 ∇2g(qε, sε)
.
However, the algebraic complexity is quite more challenging in the non-commutative case.
Following the calculation in the comparison arguments in [17] (and also [16, 95]), with the
help of a computer algebra system, we get that there exists C > 0 such that
〈(M1 + λM21 )wpε ⊕ wqε , wpε ⊕ wqε〉 ≤
C
ε|w|2(zpε − zqε −
1
2xpεyqε +
1
2xqεypε)
2 (6.3.14)
82
for any λ > 0 small. We next follow the strategy in the Euclidean case from [11]. With the
help of a computer algebra system, we simplify the left hand side of the following inequalities
and obtain a constant Cβ > 0 depending only on β, such that
〈M2(wpε ⊕ wqε), (wpε ⊕ wqε)〉 ≤1
ε|w|2CβK(pε, qε, tε, sε), (6.3.15)
〈M1M2(wpε ⊕ wqε), (wpε ⊕ wqε)〉
≤ 1
ε|w|2CβK(pε, qε, tε, sε)
∣∣∣∣zpε − zqε − 1
2xpεyqε +
1
2xqεypε
∣∣∣∣ (6.3.16)
and
〈M22 (wpε ⊕ wqε), (wpε ⊕ wqε)〉 ≤ |w|2CβK2(pε, qε, tε, sε). (6.3.17)
We remark that the existence of Cβ here is essentially due to the boundedness of ∇H〈p〉
and ∇2H〈p〉 in H.
By (6.3.11) and (6.3.14), we may take λ > 0 sufficiently small, depending on the size of
ε, p, t, and β, such that
〈Xw,w〉 − 〈Y w,w〉
≤ Cσ
ε|w|2(zpε − zqε −
1
2xpεyqε +
1
2xqεypε)
2 + 2σ|w|2CβK(pε, qε, tε, sε).(6.3.18)
We next apply the definition of viscosity sub- and supersolutions and get
a1 − tr(AX) + f(pε, ξ + η1) ≤ 0 (6.3.19)
and
a2 − tr(AY ) + f(qε, ξ + η2) ≥ 0. (6.3.20)
By subtracting (6.3.20) from (6.3.19), we have
a1 − a2 ≤ tr(AX)− tr(AY ) + f(qε, ξ + η2)− f(pε, ξ + η1),
which yields, by (6.3.18) and (A1),
a1 − a2 ≤Cσ
ε‖A‖(zpε − zqε −
1
2xpεyqε +
1
2xqεypε)
2 + L2(ρ)|pε · q−1ε |
+ (2σCβ‖A‖+ 2βµσL1)K(pε, qε, tε, sε)
(6.3.21)
83
with ρ = |p|+ 1 for ε > 0 sufficiently small.
Since we have (6.3.9), we now can take ε > 0 small to get
C
ε(zpε − zqε −
1
2xpεyqε +
1
2xqεypε)
2 + L2(ρ)|pε · q−1ε | ≤
σ
T 2.
Taking λ > 0 accordingly small and
α > 2Cβ‖A‖+ βµLf ,
we reach a contradiction to (6.3.10).
An immediate consequence is certainly the uniqueness of solutions with at most expo-
nential growth at space infinity.
Corollary 6.3.2 (Uniqueness of solutions). Assume that (A1) and (A2) hold. Let u0 ∈
C(H). Then there is at most one continuous viscosity solution u of (6.0.1)–(6.0.2) satisfying
the following exponential growth condition at infinity:
(G) For any T > 0, there exists k > 0 and CT > 0 such that |u(p, t)| ≤ CT ek〈p〉 for all
(p, t) ∈ H× [0, T ].
.
The existence of viscosity solutions of (6.0.1)–(6.0.2) is not the main topic of this chapter,
but we remark that it is possible to adapt Perron’s method [27] to our current case in the
Heisenberg group, under various extra assumptions on the function f . For example, one may
further assume on (6.0.1) that
(A3) |f(p, ξ)| ≤ Cf (1 + |ξ|) for some Cf > 0 and all p ∈ H, ξ ∈ R2.
In this case, it is not difficult to verify by computation that u = Cg(p, t) + Cf t and u =
−Cg(p, t)− Cf t are respectively a supersolution and a subsolution of (6.0.1) for any C > 0
and β > 0 when α > 0 is sufficiently large. Indeed, we have
ut = Cαg + Cf ,
| tr(A(∇2Hu)∗)| ≤ C‖A‖β2µ2g,
84
and
|f(p,∇Hu)| ≤ CCfβµg + Cf ,
where µ is the same constant as in the proof of Theorem 6.3.1. Therefore, by (A3), we get
ut − tr(A(∇2Hu)∗) + f(p,∇Hu) ≥ 0
when α > ‖A‖β2µ2 + Cfβµ. The verification for u is similar.
If there exist C > 0 and k > 0 such that
−Cek〈p〉 ≤ u0(p) ≤ Cek〈p〉 for all p ∈ H, (6.3.22)
then classical arguments [27] show that the supremum over all subsolutions bounded by u
and u is in fact a unique continuous solution. We state the result below without more details
in its proof.
Corollary 6.3.3. Assume the Lipschitz conditions (A1), (A2) and the growth condition
(A3). Let u0 ∈ C(H) satisfy (6.3.22) for some C > 0 and k > 0. Then there exists a unique
continuous solution u of (6.0.1)–(6.0.2) satisfying the exponential growth condition (G).
6.4 LIPSCHITZ PRESERVING PROPERTIES
In the Euclidean space, Lipschitz continuity preserving of the solutions to some partial
differential equations is a very important property: when the initial value u0 is Lipschitz
continuous , the unique solution u(x, t) is Lipschitz continuous in x as well for any t ≥ 0.
In what follows, assuming appropriate growth conditions for the initial value u0 and
its derivatives, we sketch a proof of these properties for the unique smooth solution of the
classical heat equation:
ut −∆u = 0 in Rn × (0,∞), (6.4.1)
with u(·, 0) = u0(·) in Rn, where ∆u denotes the usual (Euclidean) Laplacian operator acted
on u.
85
By differentiating the equation with respect to the space variables, one may easily see
that each of the components of∇u satisfies the heat equation (6.4.1), which, by the maximum
principle, implies that ∇u(·, t) is bounded for any t ≥ 0 if ∇u0 is bounded in Rn.
We intend to extend these preserving properties to nonlinear equations in the Heisenberg
group H. Notions and properties of Lipschitz continuity and convexity in the Heisenberg
group are available in the literature [30, 91, 70]. Recall that a function u is said to be
Lipschitz continuous in H if there exists L > 0 such that
|u(p)− u(q)| ≤ LdL(p, q)
for all p, q ∈ H. It is clear that Lipschitz continuity is left invariant.
As observed above, besides necessary applications of a comparison principle, the key in
the straightforward proofs for the Euclidean case lies at differentiating the equation and
interchanging derivatives. This is however not applicable directly in the Heisenberg group,
since the mixed second derivatives in the Heisenberg group are not commutative in general.
In fact, our counterexamples show that preserving of Lipschitz continuity fail even for very
simple linear equations; see Examples 6.4.1.
On the other hand, there are many examples on Lipschitz preserving in the Heisenberg
group. We have seen from the previous sections that there are some sufficient conditions to
guarantee the equivalence between Lipschitz continuity of a function with respect to both
metrics dL and dR; see Definition 6.1.4, Proposition 6.1.5 and Proposition 6.5.6.
We thus can obtain the Lipschitz continuity properties by first investigating them with
respect to the right invariant metric dR and then using the additional assumptions to get
the Lipschitz continuity with respect to dL.
In this section, we strengthen the assumption (A2) on f ; we assume
(A2’) the function f(p, ξ) is globally Lipschitz continuous in p with respect to the metric dR,
i.e., there exists L0 > 0 such that
|f(p, ξ)− f(q, ξ)| ≤ L0|p · q−1|G (6.4.2)
for all p, q ∈ H and ξ ∈ R2.
86
6.4.1 Right invariant Lipschitz continuity preserving
We first discuss the Lipschitz continuity based on the standard gauge metric dL (or equiv-
alently, the Carnot-Caratheodory metric). It turns out that even the simplest first order
linear equation will not preserve such Lipschitz continuity.
Example 6.4.1. Fix h0 = (1, 1) ∈ R2. Let us consider the equation
ut − 〈h0,∇Hu〉 = 0 in H
with u(p, 0) = u0(p) = |p|G for p ∈ H. By direct verification and Corollary 6.3.2, the unique
solution is
u(p, t) = |p · ht|G = dR(p, h−1t),
where h = (1, 1, 0) ∈ H0. However, it is not Lipschitz continuous with respect to dL. Indeed,
similar to Example 6.1.3, one may choose p1 = (−t − ε,−t + ε,−εt) and p2 = tv−1 =
(−t,−t, 0), which gives
u(p1, t)− u(p2, t) = |p1 · p−12 |G = (4ε4 + 64ε2t2)
14
but
dL(p1, p2) = |p−11 · p2|G =
√2ε.
The example above directs us to first consider the Lipschitz continuity with respect to
dR. The following result is an immediate consequence of Theorem 6.3.1.
Theorem 6.4.2 (Preserving of right invariant Lipschitz continuity). Assume that f : H ×
R2 → R satisfies the assumptions (A1), (A2’) and (A3). Let u ∈ C(H × [0,∞)) be the
unique solution of (6.0.1)–(6.0.2) satisfying the growth condition (G). If there exists L > 0
such that
|u0(p)− u0(q)| ≤ LdR(p, q) (6.4.3)
for all p, q ∈ H, then
|u(p, t)− u(q, t)| ≤ (L+ L0t)dR(p, q) (6.4.4)
87
for all p, q ∈ H and t ≥ 0. In particular, there exists Cρ > 0 depending on ρ > 0 and t ≥ 0
such that
|u(p, t)− u(q, t)| ≤ CρdL(p, q)12 (6.4.5)
for all p, q ∈ H with |p|, |q| ≤ ρ. Moreover, when f does not depend on the space variable p,
(6.4.4) holds with L0 = 0.
Proof. By symmetry, we only need to prove that
u(p, t)− u(h−1 · p, t) ≤ (L+ L0t)|h|G (6.4.6)
for all p, h ∈ H and t ≥ 0. It suffices to show that
v(p, t) = u(h−1 · p, t) + (L+ L0t)|h|G
is a supersolution of (6.0.1)–(6.0.2) for any h ∈ H. To this end, we recall the left invariance
of horizontal derivatives in the Heisenberg group, which implies that v is a supersolution of
vt − tr(A∇2Hv) + f(h−1 · p,∇Hv) = L0|h|G in H× (0,∞).
Since
|f(h−1 · p,∇Hv)− f(p,∇Hv)| ≤ L0|h|G
due to (6.4.2), we easily see that v is a supersolution of (6.0.1). Also, by (6.4.3), we have
u(p, 0) ≤ v(p, 0) for all p ∈ H. We conclude the proof of (6.4.6) by applying Theorem 6.3.1.
The Holder continuity (6.4.5) follows from Proposition 6.1.2.
In view of Proposition 6.1.5, we may use the theorem above to show the preserving
of Lipschitz continuity in the standard gauge metric under the assumption of evenness or
vertical evenness.
88
Corollary 6.4.3 (Lipschitz preserving of even solutions). Assume that f : H × R2 → R
satisfies the conditions (A1), (A2’) and (A3). Let u ∈ C(H× [0,∞)) be the unique solution
of (6.0.1)–(6.0.2) satisfying the growth condition (G). Assume also that u(·, t) is an even or
vertically even function. If there exists L > 0 such that
|u0(p)− u0(q)| ≤ LdL(p, q) (6.4.7)
for all p, q ∈ H, then
|u(p, t)− u(q, t)| ≤ (L+ L0t)dL(p, q) (6.4.8)
for all p, q ∈ H and t ≥ 0. In particular, when f does not depend on the space variable p,
then (6.4.8) holds with L0 = 0.
6.4.2 A special class of Hamilton-Jacobi equations
For the case of first order Hamilton-Jacobi equations (A = 0), if in addition we assume that
f : R2 → R is in the form that f(ξ) = m(|ξ|) with m : R→ R locally uniformly continuous,
then the Lipschitz preserving property of a bounded solution can be directly shown without
the evenness assumption. More precisely, we study equations in the form of
ut +m(|∇Hu|) = 0 in H× (0,∞), (6.4.9)
where m : R→ R is a locally uniformly continuous function, with initial condition u(·, 0) =
u0(·) bounded Lipschitz continuous with respect to dL in H. Since the assumption on m is
quite weak, our uniqueness and existence results for unbounded solutions in Section 6.3 do
not apply.
For solutions bounded in space, see [98] for a uniqueness theorem and a Hopf-Lax formula
when the Hamiltonian ξ 7→ m(|ξ|) is assumed to be convex and coercive. For instance, when
m(|ξ|) = |ξ|2/2, the unique solution of (6.4.9) can be expressed as
u(p, t) = infq∈H
t
2d2CC
(0,
(q−1 · pt
))+ u0(q)
. (6.4.10)
The Lipschitz preserving property (with respect to dL or dCC) was left as an open question
in [98]; see also [8] for a related open question but for more general Hamiltonians. In contrast
89
to the Euclidean case, it is not obvious how to prove the Lipschitz continuity by using the
Hopf-Lax formula (6.4.10). We here give an answer to this question using a PDE approach.
Theorem 6.4.4 (Lipschitz preserving for special Hamilton-Jacobi equations). Suppose that
m : R→ R is locally uniformly continuous. Let u be the unique viscosity solution of (6.4.9)
with u(·, 0) = u0(·) bounded in H. If u0 is Lipchitz with respect to dL in H, i.e., there exists
L > 0 such that (6.4.7) holds for any p, q ∈ H, then for all t ≥ 0
|u(p, t)− u(q, t)| ≤ LdL(p, q)
for all p, q ∈ H.
Proof. Under the assumptions above, it is known [98] that for any fixed T > 0, there is a
unique bounded continuous viscosity solution in H× [0, T ). We only need to show that
u(p, t)− u(q, t) ≤ LdL(p, q)
for all p, q ∈ H and t ∈ [0, T ). The other part can be shown by a symmetric argument.
By Young’s inequality applied to (6.4.7), we obtain
u0(p)− u0(q) ≤ LdL(p, q)4
4δ4+
3Lδ43
4(6.4.11)
for all δ > 0 and p, q ∈ H. It then suffices to show that
u(p, t)− u(q, t) ≤ LdL(p, q)4
4δ4+
3Lδ43
4(6.4.12)
for all δ > 0 and p, q ∈ H. To this end, we fix δ > 0 and prove below that
uL(p, t) = infq∈H
u(q, t) + CdL(p, q)4
with C = L/4δ4 is a supersolution of (6.4.9). Suppose there exist a bounded open set
O ⊂ H× (0, T ), φ ∈ C2(O) and (p, t) ∈ O such that
(uL − φ)(p, t) < (uL − φ)(p, t)
90
for all (p, t) ∈ O. We may also assume that φ(p, t)→ −∞ when (p, t)→ ∂O. Then for any
ε > 0 sufficiently small,
Φε(p, q, t, s) = u(q, t) + CdL(p, q)4 − φ(p, s) +(t− s)2
ε
attains a minimum at (pε, qε, tε, sε) ∈ H×H× [0,∞)× [0,∞). A standard argument yields
pε, qε → p and tε, sε → t as ε→ 0, which, in particular, implies that tε, sε 6= 0. The minimum
also implies that
∇Hφ1(pε) = ∇Hφ(pε, sε) and φt(pε, sε) =2(tε − sε)
ε, (6.4.13)
where φ1(p) = CdL(p, qε)4.
We next apply the definition of supersolutions and get
a+m(|∇Hφ2(qε)|) ≥ 0, (6.4.14)
where
a =2(tε − sε)
εand φ2(q) = −CdL(pε, q)
4.
By (6.4.13), in order to prove that uL is a supersolution, we only need to substitute ∇Hφ2(qε)
in (6.4.14) with ∇Hφ1(pε). By direct calculation, we have
∇pHdL(p, q)4 = 4
(δ1(δ2
1 + δ22)− 4δ2δ3, δ2(δ2
1 + δ22) + 4δ1δ3
)and
∇qHdL(p, q)4 = 4
(− δ1(δ2
1 + δ22)− 4δ2δ3, −δ2(δ2
1 + δ22) + 4δ1δ3
)with p = (xp, yp, zp), q = (xq, yq, zq) and
δ1 = xp − xq, δ2 = yp − yq, δ3 = zp − zq +1
2xpyq −
1
2xqyp,
This reveals that ∇pHdL(p, q)4 6= −∇q
HdL(p, q)4 and therefore ∇Hφ1(pε) 6= ∇Hφ2(qε) in gen-
eral; see [36] for more details on this aspect. However, their norms stay the same, i.e.,
|∇pHdL(p, q)4| = |∇q
HdL(p, q)4|, which turns out to be a key ingredient in this proof. In fact,
we have
|∇pHdL(p, q)4| = |∇q
HdL(p, q)4| = 4dL(p, q)2(δ21 + δ2
2)12 ,
91
which implies that |∇Hφ1(pε)| = |∇Hφ2(qε)| and their boundedness uniformly in ε. Hence,
due to (6.4.13), the equation (6.4.14) is now rewritten as
φt(pε, sε) +m(|∇Hφ(pε, sε)|) ≥ 0.
By sending ε → 0 and using the continuity of m, we conclude the verification that uL is a
supersolution. It follows that vL = uL + 3Lδ43/4 is also a supersolution of (6.4.9). Thanks
to (6.4.11), we have u(p, 0) ≤ vL(p, 0), which implies (6.4.12) by Theorem 6.3.1.
6.5 CONVEXITY PRESERVING PROPERTIES
It is well known that the convexity preserving property holds for a large class of fully nonlinear
equations in the Euclidean space [39]. The definition reads as follows: when the initial value
u0 is convex, the unique solution u(x, t) is convex in x as well for any t ≥ 0. Concerning
convexity in the Heisenberg group, the notion of h-convexity (and equivalently v-convexity)
turns out to be a natural extension of the Euclidean version. However, we cannot expect
such convexity to be preserved in general. In fact, h-convexity is not preserved even for the
first order linear equation.
Example 6.5.1 (Linear first order equations). We again consider the linear equation
ut − 〈h0,∇Hu〉 = 0 in H (6.5.1)
with h0 = (1, 1) and u(x, y, z, 0) = f(x, y, z) with f defined as in (6.1.7) for all (x, y, z) ∈ H.
Let h = (1, 1, 0) ∈ H0. As verified in Example 6.1.8, u(·, 0) is h-convex in H. However, the
unique solution
u(p, t) = f(p · ht) = (x+ t)2(y + t)2 + 2
(z +
1
2xt− 1
2yt
)2
(6.5.2)
is not h-convex for any t > 0. In fact, the symmetrized Hessian is given by
(∇2Hu)∗(p, t) =
2(y + t)2 + (y − t)2 4(x+ t)(y + t)− (x− t)(y − t)
4(x+ t)(y + t)− (x− t)(y − t) 2(x+ t)2 + (x− t)2
.
92
It is therefore easily seen that
(∇2Hu)∗(t, t, 0, t) =
8t2 16t2
16t2 8t2
,
which shows that u(·, t) is not h-convex around the point p = (t, t, 0) ∈ H for any t > 0.
The loss of convexity preserving is due to the non commutativity of the Heisenberg group
product. Although the h-convexity of a function is preserved under left translations, it is not
necessarily preserved under right translations, as indicated in Example 6.5.1. We therefore
consider right invariant h-convexity first and get left invariant h-convexity preserving in some
special cases.
Finally, our study of the convexity preserving property in the Heisenberg group is also
inspired by recent works on horizontal mean curvature flow in sub-Riemannian manifolds
[20, 36]. The mean curvature flow in Rn is known to preserve convexity [62], but it is not
clear if such a property also holds in H in general. Our analysis about convexity is only for
the simpler equation (6.0.1). However, an explicit solution of the mean curvature flow in H
that does preserve convexity can be found in Example 6.6.3; see also [36].
6.5.1 Right invariant h-convexity preserving
Definition 6.5.2 (Right invariant h-convexity). Let Ω be an open set in H and u : Ω→ R be
an upper semicontinuous function. The function u is said to be right invariant horizontally
convex or right h-convex in Ω, if for every p ∈ H and h ∈ H0 such that [h−1 · p, h · p] ⊂ Ω,
we have
u(h−1 · p) + u(h · p) ≥ 2u(p). (6.5.3)
Following the proof of Theorem 6.3.1, we can show a convexity maximum principle and
get the theorem below.
Theorem 6.5.3 (Preservation of right invariant h-convexity). Suppose that the assumptions
(A1), (A2) and (A3) hold. Let u ∈ C(H× [0,∞)) be the unique viscosity solution of (6.0.1)–
(6.0.2) satisfying the growth condition (G). Assume in addition that f is right invariant
93
concave in H× R2, i.e.,
f(h−1 · p, ξ) + f(h · p, η) ≤ 2f
(p,
1
2(ξ + η)
)(6.5.4)
for all p ∈ H, h ∈ H0 and ξ, η ∈ R2. If u0 is right invariant h-convex in H, then so is u(·, t)
for all t ≥ 0.
Proof of Theorem 6.5.3. By definition, we aim to show that
u(h−1 · p, t) + u(h · p, t) ≥ 2u(p, t)
for any p ∈ H, h ∈ H0, t ≥ 0. We assume by contradiction that there exist (p0, h0, t0) ∈
H×H0 × [0,∞) such that
u(h−10 · p0, t0) + u(h0 · p0, t0) < 2u(p0, t0).
Then there exists a positive maximizer (p, h, t) ∈ H×H0 × [0, T ) of
2u(p, t)− u(h−1 · p, t)− u(h · p, t)− 3σg(p, t)− σ
m− |h|4G− σ
T − t
with some constants m > |h|4G, T > t and σ > 0 small. Here g(p, t) = eαt+β〈p〉 with α > 0 to
be determined later and any fixed β > k. We next consider
Φ(p, q, r, h, t, s, τ) = 2u(r, τ)− u(h−1 · p, t)− u(h · q, s)− σΨε(p, q, r, t, s, τ)
−ψε(t, s, τ)− σ
m− |h|4G− σ
T − τ,
where
ψε(t, s, τ) =(t− s)2
ε+
(t− τ)2
ε+
(s− τ)2
ε,
Ψε(p, q, r, t, s, τ) = φε(p, q, r) +K(p, q, r, t, s, τ)
with
φε(p, q, r) =|p · r−1|4
ε+|q · r−1|4
ε
and
K(p, q, r, t, s, τ) = g(r, τ) + g(p, t) + g(q, s).
94
It follows that Φ has a maximizer (pε, qε, rε, hε, tε, sε, τε). As before, we denote
pε = (xpε , ypε , zpε), qε = (xqε , yqε , zqε), rε = (xrε , yrε , zrε).
Due to the penalization at space infinity, we have φε(pε, qε, rε) bounded from above
uniformly in ε. By a standard argument, we can show that there exists p ∈ H, h ∈ H0 and
t ∈ [0, T ) such that, up to a subsequence,
pε, qε, rε → p, hε → h, tε, sε, τε → t (6.5.5)
and
φε(pε, qε, rε)→ 0 (6.5.6)
as ε→ 0.
Since u0 is right invariant h-convex, we have t > 0 and therefore tε, sε, τε > 0 when ε is
sufficiently small.
Denote u−(p, t) = u(h−1ε ·p, t) and u+(p, t) = u(hε ·p, t). We now apply the Crandall-Ishii
lemma in the Heisenberg group and get, for any λ ∈ (0, 1),
(a1, ζ1, X1) ∈J2,−u−(pε, tε), (a2, ζ2, X2) ∈ J2,−
u+(qε, sε),
(a3, ζ3, X3) ∈ J2,+u(rε, τε)
such that
2a3 − a1 − a2 =σ
(T − τε)2+ σαK(pε, qε, rε, tε, sε, τε), (6.5.7)
the horizontal projections of ζi can be expressed as ξi + ηi (i = 1, 2, 3) with
−ξ1 = σ∇pHφε(pε, qε, rε), −ξ2 = σ∇q
Hφε(pε, qε, rε), 2ξ3 = σ∇rHφε(pε, qε, rε),
−η1 = σ∇Hg(pε, tε), −η2 = σ∇Hg(qε, sε), 2η3 = σ∇Hg(rε, τε),
and
〈(2X3 −X1 −X2)w,w〉 ≤ 〈(σM + λσ2M2)wpε ⊕ wqε ⊕ wrε , wpε ⊕ wqε ⊕ wrε〉, (6.5.8)
95
for all w ∈ R2, where M = ∇2Ψε(pε, qε, rε, hε, tε, sε, τε) is a 9× 9 symmetric matrix, wpε , wqε
are respectively taken as in (6.3.12) and (6.3.13), and
wrε =
(w1, w2,
1
2w2xrε −
1
2w1yrε
).
By calculation, it is easily seen that
2ξ3 = ξ1 + ξ2. (6.5.9)
We next set
M1 = ∇2φε(pε, qε, rε)
and
M2 = ∇2K(pε, qε, rε, tε, sε, τε)
Then M = M1 + M2. To investigate the right hand side of (6.5.8), we first give estimates
the terms involving M1, which is a variant of (6.3.14) for three space variables. Note that
M1 = M ′1 +M ′′
1 ,
where
M ′1 =
1
ε∇2|p · r−1|4, M ′′
1 =1
ε∇2|q · r−1|4.
By direct calculations, we get
〈M ′1wpε ⊕ wqε ⊕ wrε , wpε ⊕ wqε ⊕ wrε〉 = 0,
〈M ′′1wpε ⊕ wqε ⊕ wrε , wpε ⊕ wqε ⊕ wrε〉 = 0
and
〈M ′21 wpε ⊕ wqε ⊕ wrε , wpε ⊕ wqε ⊕ wrε〉 =
512
εm2
1|w|2,
〈M ′′21 wpε ⊕ wqε ⊕ wrε , wpε ⊕ wqε ⊕ wrε〉 =
512
εm2
2|w|2,
〈M ′1M
′′1wpε ⊕ wqε ⊕ wrε , wpε ⊕ wqε ⊕ wrε〉 =
256
εm1m2|w|2,
where
m1 = zpε − zrε +1
2ypεxrε −
1
2xpεyrε , m2 = zqε − zrε +
1
2yqεxrε −
1
2xqεyrε .
96
It follows that
〈(M1 + λM21 )wpε ⊕ wqε ⊕ wrε , wpε ⊕ wqε ⊕ wrε〉 ≤
512λ
ε(m2
1 +m22 +m1m2)|w|2, (6.5.10)
which implies that
〈(M1 + λM21 )wpε ⊕ wqε ⊕ wrε , wpε ⊕ wqε ⊕ wrε〉 ≤
C
ε|w|2(|pε · r−1
ε |4 + |qε · r−1ε |4) (6.5.11)
for some C > 0 independent of ε and λ. On the other hand, with the help of computer
algebra system, we obtain estimates similar to (6.3.15), (6.3.16) and (6.3.17). In fact, we get
a constant Cβ such that, when λ > 0 is small enough (depending on ε),
〈M2(wpε ⊕ wqε ⊕ wrε), (wpε ⊕ wqε ⊕ wrε)〉 ≤ Cβ|w|2K(pε, qε, rε, tε, sε, τε) (6.5.12)
〈λ(M1M2 +M2M1 +M22 )(wpε ⊕ wqε ⊕ wrε), (wpε ⊕ wqε ⊕ wrε)〉
≤ 2Cβ|w|2K(pε, qε, rε, tε, sε, τε)(6.5.13)
for some constant µ > 0 independent of ε, β and σ and satisfying (6.3.7). As remarked in
the proof of Theorem 6.3.1, we obtain the constant Cβ thanks to the boundedness of ∇H〈p〉
and ∇2H〈p〉 in H.
Combining (6.5.8), (6.5.11) and (6.5.13), we have
〈(2X3 −X1 −X2)w,w〉 ≤ Cσ
ε‖w‖2(|pε · r−1
ε |4 + |qε · r−1ε |4)
+ 2σ‖w‖2CβK(pε, qε, rε, tε, sε, τε).
(6.5.14)
when λ > 0 and σ > 0 are sufficiently small.
Since the horizontal derivatives are left translation invariant, the functions u− and u+
are respectively solutions of
(u−)t − tr(A∇2Hu−) + f(h−1
ε · p,∇Hu−) = 0 in H× (0,∞)
and
(u+)t − tr(A∇2Hu+) + f(hε · p,∇Hu+) = 0 in H× (0,∞).
97
Applying the definition of viscosity subsolutions and supersolution, we have
a1 − tr(AX1) + f(h−1ε · pε, ξ1 + η1) ≥ 0, (6.5.15)
a2 − tr(AX2) + f(hε · qε, ξ2 + η2) ≥ 0, (6.5.16)
a3 − tr(AX3) + f(rε, ξ3 + η3) ≤ 0. (6.5.17)
Subtracting (6.5.15) and (6.5.16) from twice (6.5.17), we get
2a3 − a1 − a2 ≤ trA(2X3 −X1 −X2) + E, (6.5.18)
where
E = f(h−1ε · pε, ξ1 + η1) + f(hε · qε, ξ2 + η2)− 2f(rε, ξ3 + η3)
It follows from the concavity assumption (6.5.4), the relation (6.5.9) and (A1)-(A2) that
E ≤ f(h−1ε · pε, ξ1 + η1)− f(h−1
ε · rε, ξ1 + η1) + f(hε · qε, ξ2 + η2)− f(hε · rε, ξ2 + η2)
+ f(h−1ε · rε, ξ1 + η1) + f(hε · rε, ξ2 + η2)− 2f(rε,
1
2(ξ1 + ξ2 + η1 + η2))
+ 2f(rε,1
2(ξ1 + ξ2 + η1 + η2))− 2f(rε, ξ3 + η3)
≤ LR(|h−1ε · rε · p−1
ε · h|+ |h−1ε · rε · q−1
ε · hε|) + Lf |η1 + η2 − 2η3|(6.5.19)
with R = (|p|+ 1) and ε > 0 small. Also, by (6.3.7), we have
|η1 + η2 − 2η3| ≤ 2(|η1|+ |η2|+ |η3|) = 2σβµK(pε, qε, rε, tε, sε, τε).
In view of (6.5.14), (6.5.18) and (6.5.19), we then obtain
2a3 − a1 − a2
≤Cσε
(|pε · r−1ε |4 + |qε · r−1
ε |4) + LR|h−1ε · rε · p−1
ε · h|+ LR|h−1ε · rε · q−1
ε · hε|
+ 2σ(Cβ‖A‖+ Lfβµ)K(pε, qε, rε, tε, sε, τε).
(6.5.20)
In view of (6.5.5) and (6.5.6), we can take ε > 0 small such that
Cσ
ε(|pε · r−1
ε |4 + |qε · r−1ε |4) + LR|h−1
ε · rε · p−1ε · h|+ LR|h−1
ε · rε · q−1ε · hε| <
σ
T 2,
98
which, by (6.5.20), implies
2a3 − a1 − a2 ≤σ
T 2+ 2σ(Cβ‖A‖+ Lfβµ)K(pε, qε, rε, tε, sε, τε).
It clearly contradicts (6.5.7) when α is chosen to satisfy
α > 2‖A‖Cβ + 2Lfβµ.
One may further generalize this result for (6.0.3) by assuming that F is concave in all
arguments.
Remark 6.5.4. The concavity assumption (6.5.4) on the operator f is stronger than the as-
sumptions of the convexity results in the Euclidean space as shown in [39, 69]. In particular,
the concavity of ξ 7→ f(p, ξ) is not needed in the Euclidean case. We here need this assump-
tion, since there are no expressions of h-convexity in H corresponding to the following one
for the Euclidean convexity
u(ξ) + u(η) ≥ 2u
(ξ + η
2
)for all ξ, η ∈ Rn. It is not clear to us whether the assumption (6.5.4) can be weakened.
Example 6.5.5. Let us revisit Example 6.5.1. Since the equation (6.5.1) and the solution
(6.5.2) satisfy all of the assumptions in Theorem 6.5.3, the right invariant h-convexity of the
solution is preserved, though the h-convexity is not. Indeed, if u(p, t) is given by (6.5.2),
then by direct calculation we obtain, for all p = (x, y, z), h = (h1, h2, 0) and t ≥ 0,
u(h · p, t) + u(h−1 · p, t)
= (x+ h1 + t)2(y + h2 + t)2 + 2
(z +
1
2h1y −
1
2h2x+
1
2(x+ h1)t− 1
2(y + h2)t
)2
= 2(x+ t)2(y + t)2 + 4
(z +
1
2xt− 1
2yt
)2
+ (h1(y + t)− h2(x+ t))2 + 2h21(y + t)2
+ 2h22(x+ t)2 + 8(x+ t)(y + t)h1h2 + 2h2
1h22
= 2(x+ t)2(y + t)2 + 4
(z +
1
2xt− 1
2yt
)2
+ 3(h1(y + t) + h2(y + t))2 + 2h21h
22
≥ 2(x+ t)2(y + t)2 + 4
(z +
1
2xt− 1
2yt
)2
= 2u(p, t).
99
6.5.2 Left invariant h-convexity preserving
We next discuss some special cases, where h-convexity and right invariant h-convexity are
equivalent.
Proposition 6.5.6 (Evenness). Let u be an even or vertically even function on H . Then
u is h-convex in H if and only if u is right invariant h-convex in H.
Proof. By definition, u is h-convex if u satisfies (6.1.5) for any p ∈ H and h ∈ H0. Since u is
even, it is easily seen that (6.1.5) holds if and only if
u(h · p) + u(h−1 · p) ≥ u(p),
where p is given as in (6.1.4), or
u(h · p−1) + u(h−1 · p−1) ≥ u(p−1)
for all p ∈ H and h ∈ H0, which is equivalent to saying
u(h · p) + u(h−1 · p) ≥ u(p) for all p ∈ H and h ∈ H0.
Another sufficient condition for equivalence between the h-convexity and the left h-
convexity of a function u on H is that u has a separate structure; namely,
u(x, y, z) = f(x, y) + g(z) (6.5.21)
for any (x, y, z) ∈ H.
Proposition 6.5.7 (Separability). Let u be a function on H with a separate structure as in
(6.5.21). Then u is h-convex in H if and only if u is right invariant h-convex in H.
100
Proof. Suppose u can be written as in (6.5.21). Setting p = (x, y, z) and h = (h1, h2), we
then have
u(p · h) = f(x+ h1, y + h2) + g(z +1
2xh2 −
1
2yh1);
u(p · h−1) = f(x− h1, y − h2) + g(z − 1
2xh2 +
1
2yh1);
u(h · p) = f(x+ h1, y + h2) + g(z +1
2yh1 −
1
2xh2);
u(h−1 · p) = f(x− h1, y − h2) + g(z − 1
2yh2 +
1
2xh2).
It is easily seen that in this case
u(p · h−1) + u(p · h) = u(h−1 · p) + u(h · p),
which immediately yields the equivalence of (6.1.5) and (6.5.3) in H.
The following result on preserving of the h-convexity itself is an immediate consequence
of Theorem 6.5.3, Propositions 6.5.6 and 6.5.7.
Corollary 6.5.8 (H-convexity preserving under evenness or separability). Assume that f
satisfies (A1)–(A3) and the concavity condition (6.5.4) for all p ∈ H, h ∈ H0 and ξ, η ∈ R2.
Let u ∈ C(H× [0,∞)) be the unique viscosity solution of (6.0.1)–(6.0.2) satisfying the growth
condition (G). Assume in addition that for any t ≥ 0, u(·, t) either is an even or vertically
even function or has a separable structure as in (6.5.21). If u0 is h-convex in H, then so is
u(·, t) in H for all t ≥ 0.
6.6 MORE EXAMPLES
In this section, we provide more examples, where the h-convexity is preserved.
Example 6.6.1. Let u0(x, y, z) = (x2 + y2)2 − 8z2. It is not difficult to see that u0 is an
h-convex function in H. Consider the heat equation
ut −∆Hu = 0 in H× (0,∞) (6.6.1)
101
with u(·, 0) = u0 in H, where ∆H denotes the horizontal Laplacian operator in the Heisenberg
group, i.e., ∆Hu = tr(∇2Hu)∗. The unique solution of (6.6.1) in this case is
u(x, y, z, t) = (x2 + y2)2 − 8z2 + 12(x2 + y2)t+ 24t2 (6.6.2)
for all (x, y, z) ∈ H and t ≥ 0 and it actually preserves the h-convexity of the initial value
u0.
Example 6.6.2. The solution as in (6.6.2) looks special, since it can be written as the sum of
a function of x, y, t and a function of z. A more complicated solution of the heat equation
(6.6.1) is
u(x, y, z, t) = (x2 + y2)z2 +1
24(x2 + y2)3 + (4z2 + 2(x2 + y2)2)t+ 17(x2 + y2)t2 +
68
3t3 (6.6.3)
which contains mixed terms of x, y and z. By direct calculation, one can also show that
u(·, t) satisfies (6.1.6) in H in the classical sense for everywhere t ≥ 0.
Example 6.6.3. We recall another example in [36] for the level-set mean curvature flow
equation in H. The equation is of the form
ut − |∇Hu| divH
(∇Hu
|∇Hu|
)= 0 in H× (0,∞), (6.6.4)
where divH stands for the horizontal divergence operator in the Heisenberg group. An
explicit solution is
u(x, y, z, t) = (x2 + y2)2 + 16z2 + 12(x2 + y2)t+ 12t2.
This is also an example of h-convexity preserving but unfortunately is not covered by our
current results.
102
7.0 A GAME-THEORETIC PROOF OF CONVEXITY PRESERVATION
OF SOME NONLINEAR DIFFERENTIAL EQUATIONS
Among many properties of the viscosity solutions, the convexity preserving property is known
to be important for various linear and nonlinear parabolic equations arising in geometry,
material sciences and image processing. This property of viscosity solutions was proved to
hold in a very general class of degenerate parabolic equations using the so called convexity
maximum principle in the Euclidean spaces [39]. We have also seen some discussions of this
property on the Heisenberg group in the previous chapter.
The discrete game interpretations of various elliptic and parabolic PDEs ([78, 100, 101,
79, 97, 96], etc) have recently attracted great attention. The game related methods are also
used as a new tool in different contexts. We will introduce the game interpretation of the
level set mean curvature flow equation and normalized p-Laplace equations.
Based on game-theoretic interpretations, we give a simpler proof of the convexity pre-
serving of the level sets and solutions to the mean curvature flow equation in R2. Our
method also applies to normalized parabolic p-Laplace equations. Our new proofs are based
on investigating game strategies or iterated applications of Dynamic Programming Principles
(DPP), which is very different from the standard proofs in the literature. We also use this
method to study convexity preserving for the Neumann boundary problems.
7.1 CONVEXITY PRESERVING PROPERTIES
The mean curvature flow describes the motion of a surface in Rn governed by the law that
the normal velocity is equal to the mean curvature. The convexity preserving property for
103
mean curvature flow is a well-known result by Huisken [62], saying that an evolving surface
by mean curvature stays convex if the initial surface is convex; see also [37] for the two
dimensional case in detail. This property was later formulated in terms of level set method
in [34] and [39].
(MCF)
ut − |∇u| div
(∇u|∇u|
)= 0 in Rn × (0,∞), (7.1.1)
u(x, 0) = u0(x) in Rn, (7.1.2)
where u0 is assumed to be a Lipschitz continuous function in Rn.
In [34] and [39], those authors proved the following.
Theorem 7.1.1. Suppose u0 is a Lipschitz continuous function on Rn. Let u be the unique
viscosity solution of (MCF). If the set x ∈ Rn : u0(x) ≥ 0 is convex, then the set x ∈
Rn : u(x, t) ≥ 0 is also convex for any t ≥ 0.
The proof in [34] is based on a regularization of the mean curvature operator for the
corresponding stationary problem in a convex domain Ω ⊂ Rn, that is,
(SP)
−∆G1 u = −|∇u| div
(∇u|∇u|
)= 1 in Ω, (7.1.3)
u = 0 on ∂Ω, (7.1.4)
and an application of more classical convexity maximum principle by Korevaar [81] and
Kennington [75] for C2 solutions of uniformly elliptic equations. In two dimensions, Barron,
Goebel and Jensen [12] recently gave a PDE proof for the convexity of level sets for any
strict subsolution or strict supersolution of −∆G1 u = 0.
The convexity (concavity) preserving property of viscosity solutions was later studied for
a general class of degenerate parabolic equations in [39], including the mean curvature flow
equation and normalized p-Laplace equations.
(PL)
ut − tr
((I + (p− 2)
∇u⊗∇u|∇u|2
)∇2u
)= f(x) in Rn × (0,∞), (7.1.5)
u(x, 0) = u0(x) in Rn, (7.1.6)
where f : Rn → R is a given continuous function.
We list the theorem for the convexity preserving of the viscosity solutions to (MCF).
104
Theorem 7.1.2. Suppose u0 is a concave and Lipschitz continuous function on Rn. Let u
be the unique viscosity solution of (MCF). Then u(x, t) is also concave in x for any t ≥ 0.
The classical argument relies on a generalized version of convexity comparison principle;
we refer to [69] for related results. More precisely, it is shown, by adding more variables in
the usual comparison theorem, that the continuous solution u satisfies
u(x, t) + u(y, t)− 2u(z, t) ≤ |x+ y − 2z|
for any x, y, z ∈ Rn and t ≥ 0 if the same holds initially:
u0(x) + u0(y)− 2u0(z) ≤ |x+ y − 2z|.
The concavity preserving property is then a special case when z = (x + y)/2. We remark
that the convexity of solutions to various PDEs has also been extensively studied in [72, 73,
107, 31, 2] etc.
7.2 GAME INTERPRETATIONS OF VISCOSITY SOLUTIONS
7.2.1 Tug of War game and normalized parabolic p-Laplace equations
Let us first discuss a game interpretation of parabolic normalized ∞-Laplace equation.
ut − tr
(∇u⊗∇u|∇u|2
∇2u
)= 0 in Rn × (0,∞),
u(x, 0) = u0(x) in Rn.
Fix ε > 0 be the stepsize of the game. The game starts from x ∈ Rn. Repeat N(= [2t/ε2])
times the following.
(1) Toss a fair coin;
(2) The winner of the first step choose a vector v in the ball B(0, ε) (abbreviated as Bε);
(3) The game position moves from x to x+ v.
105
Fix any strategies SI of Player I and SII of Player II. Let yN denote the final stage of
this game under SI and SII . Player I (II) minimizes (maximizes) the value of u0 at yN .
Denote by uε(x, t) the value function:
uε(x, t) = infSI
supSII
E(u0(yN)),
where E stands for the expectation. This game described above is often referred to as
tug-of-war game. Dynamic Programming Principle (DPP) of this game is
uε(x, t) =1
2supv∈Bε
uε(x+ v, t− ε2
2
)+
1
2infv∈Bε
uε(x+ v, t− ε2
2
).
Following the argument in [96], one can prove uε → u locally uniformly as ε→ 0.
Then we discuss the game interpretation of normalized p-Laplace equations (PL) with
2 ≤ p ≤ ∞. Fix ε > 0 be the stepsize of the game. The game starts from x ∈ Rn. Let
σp =
1
2(p+n)if 2 ≤ p <∞,
12
if p =∞.
Repeat N(= [t/σpε2]) times the following.
(1) Toss a biased coin with probabilities α to get heads and β to get tails, α + β = 1;
(2) If they get heads, they play a tug-of-war game;
(3) If they get tails, the game state moves randomly in the ball B(xn, ε).
Fix any strategies SI of Player I and SII of Player II. Let yN denote the final stage of
this game under SI and SII . Player I (II) minimizes (maximizes) the value of u0 at yN .
Denote by uε(x, t) the value function:
uε(x, t) = infSI
supSII
ESI ,SII(u0(yN)).
This game described above is often referred to as tug-of-war game with noise.
The equation is related to the DPP below:
uε(x, t) =α
2supv∈Bε
uε(x+ v, t− σpε2) +α
2infv∈Bε
uε(x+ v, t− σpε2)
+ β
v∈Bε
uε(x+ v, t− σpε2) dv + ε2f(x).(7.2.1)
106
uε(x, t) = u0(x) for all x ∈ Rn and t < ε2. (7.2.2)
Here α ≥ 0, β ≥ 0 satisfy α+ β = 1 and are determined by the choice of p. In fact, it is
known from [96, 97] that
α =p− 2
p+ nand β =
2 + n
p+ n.
The presence of the constant σp in (7.2.1) is due to the different forms of DPP for our
convenience later; in contrast to [96], our formula does not use averages with respect to t.
One may show that uε converges, as ε→ 0, to the unique solution u of (PL) locally uniformly
in Rn × [0,∞) by following [96].
7.2.2 The deterministic game and the mean curvature flow equations
Let us first review the game proposed in [78] for motion by curvature. A marker, representing
the game position or game state, is initialized at x ∈ R2 from time 0. The maturity time
given is denoted by t. Let the step size for space be ε > 0. Time ε2 is consumed for each
step. The total game steps N can be regarded as [t/ε2].
Two players, Player I and Player II, participate in the game: Player I intends to maximize
at the final state an objective function, which in our case is u0, while Player II is to minimize
it. At each turn,
(1) Player I chooses in R2 a unit vector v;
(2) Player II has the right to keep or reverse the choice of Player I, which determines a sign
b = ±1;
(3) The marker is moved from the present state x to x+√
2εbv.
To give a mathematical description, we denote
S1 = v ∈ R2 : |v| = 1.
Then the inductive state equation writes as zk+1 = zk +√
2εbkvk, k = 0, 1, . . . , N − 1;
z0 = x,
107
where vk ∈ S1 and bk = ±1.
Hereafter, for any x ∈ Rn and s ∈ [0,∞), z(s;x) = zm stands for the game state at the
step m = [s/ε2] starting from x under the competing strategies so that our games look like
continuous ones. The value function is defined as
uε(x, t) := maxv1∈S1
minb1=±1
. . . maxvN∈S1
minbN=±1
u0(z(t;x)). (7.2.3)
It is also known [78, Appendix B] that uε preserves Lipschitz continuity of u0 in space. The
dynamic programming principle is as below:
uε(x, t) = maxv∈S1
minb=±1
uε(x+√
2εbv, t− ε2), (7.2.4)
It is employed to show the following theorem. We assume that
(A) u0 is a bounded and Lipschitz continuous function in R2 satisfying that u0 ≥ a and u0−a
has compact support Ka for some constant a ∈ R.
Theorem 7.2.1 (Theorem 1.2 in [78]). Assume that u0 satisfies (A). Let uε be the value
function defined as in (7.2.3). Then uε converges, as ε→ 0, to the unique viscosity solution
of (MCF) uniformly on compact subsets of R2 × (0,∞).
We recall the definition of viscosity solutions for (MCF) below.
Definition 7.2.2 ([38]). A locally bounded upper (resp., lower) semicontinuous function
u is called a subsolution (resp., supersolution) of (7.1.1) if for any (x0, t0) ∈ R2 and φ ∈
C2(R2× [0,∞)) such that u−φ attains a (strict) maximum (resp., minimum) at (x0, t0), we
have
φt − |∇φ| div
(∇φ|∇φ|
)≤ 0 at (x0, t0), (7.2.5)(
resp., φt − |∇φ| div
(∇φ|∇φ|
)≥ 0 at (x0, t0)
)when ∇φ(x0, t0) 6= 0 and
φt(x0, t0) ≤ 0, (resp., φt(x0, t0) ≥ 0)
when ∇φ(x0, t0) = 0 and ∇2φ(x0, t0) = O.
108
A locally bounded continuous function u is called a solution if it is both a subsolution
and a supersolution. A subsolution (resp., supersolution, solution) u of (7.1.1) is said to be
a subsolution (resp., supersolution, solution) of (MCF) if it further satisfies u(x, 0) ≤ u0(x)
(resp., u(x, 0) ≥ u0(x), u(x, 0) = u0(x)) for all x ∈ R2.
7.2.3 A modified game
Let us slightly change the rules of the game described above: we keep the basic setting of
the game and the objectives of both Players, but at each round, we now ask that
(1) Player I chooses v, v′ ∈ S1;
(2) Player II determines b = ±1 and b′ = ±1;
(3) The marker is moved from the present state x to x+√
2ε2
(bv + b′v′).
Under the new rules above, we may define the value function
wε(x, t) := maxv1,v′1∈S1
minb1,b′1=±1
. . . maxvN ,v
′N∈S1
minbN ,b
′N=±1
u0(z(t;x)), (7.2.6)
where z(t;x) = zN is the solution of the following state equation: zk+1 = zk +
√2ε
2(bkvk + b′kv
′k), k = 0, 1, . . . , N − 1;
z0 = x,
It is clear that the new dynamic programming principle is
wε(x, t) = maxv,v′∈S1
minb,b′=±1
wε
(x+
√2ε
2(bv + b′v′), t− ε2
), (7.2.7)
for any t ≥ ε2. The Lipschitz preserving property of wε can be shown by following [78,
Appendix B].
Proposition 7.2.3 (Lipschitz continuity preserving property). Let wε be the value function
associated to the modified game described above. If there exists L > 0 such that
|u0(x)− u0(y)| ≤ L|x− y| for any x, y ∈ R2,
then wε(x, t) satisfies
|wε(x, t)− wε(y, t)| ≤ L|x− y| for any x, y ∈ R2, t ≥ 0 and ε > 0. (7.2.8)
109
Proof. We prove this result by induction. By (7.2.7), we have
wε(x, ε2) = maxv,v′∈S1
minb,b′=±1
u0
(x+
√2ε
2(bv + b′v′)
)
wε(y, ε2) = maxv,v′∈S1
minb,b′=±1
u0
(y +
√2ε
2(bv + b′v′)
).
Let v0, v′0 be the maximizer in the first relation and b0, b
′0 be the minimizer (with respect to
the choice of v = v0, v′ = v′0) in the second. We have
wε(x, ε2)− wε(y, ε2) ≤ u0
(x+
√2ε
2(b0v0 + b′0v
′0)
)− u0
(y +
√2ε
2(b0v0 + b′0v
′0)
),
which, in view of the Lipschitz continuity of u0, implies that
wε(x, ε2)− wε(y, ε2) ≤ L|x− y| for all x, y ∈ R2 and ε > 0.
By applying this argument repeatedly and (7.2.7), we are led to
wε(x, t)− wε(y, t) ≤ L|x− y| for all x, y ∈ R2 and ε > 0.
Then (7.2.8) follows immediately by interchanging the roles of x and y.
We next compare this modified game with the original game. In fact, we can show that
the relaxed upper limit
w(x, t) = limsup∗ε→0
wε(x, t) = lim supε→0
wδ(y, s) : |y − x|+ |t− s| < ε, δ < ε
is a subsolution of (MCF).
Theorem 7.2.4 (Subsolution). Assume that u0 satisfies (A). Let wε be the value functions
defined as in (7.2.6). Then w as defined above is a subsolution of (7.1.1) with w(·, 0) ≤ u0
in R2.
We present a detailed proof of Theorem 7.2.4, since the game setting is quite different
from the original games in [78]. The following elementary result, whose proof can be found
in [41], is needed in our argument.
110
Lemma 7.2.5 (Lemma 4.1 in [41]). Suppose p is a unit vector in R2 and X is a real
symmetric 2× 2 matrix. Then there exists a constant M > 0 that depends only on the norm
of X, such that for any unit vector ξ ∈ R2,
|〈Xp⊥, p⊥〉 − 〈Xξ, ξ〉| ≤M |〈ξ, p〉|, (7.2.9)
where p⊥ denotes a unit orthonormal vector of p.
Remark 7.2.6. By homogeneity, when p is not necessarily a unit vector, the relation (7.2.9)
still holds for all ξ ∈ R2 provided that p⊥ is orthogonal to p and |p⊥| = |p| = |ξ|.
Proof of Theorem 7.2.4. We first show that w is a subsolution of (7.1.1). Let us assume that
there exist (x0, t0) ∈ Q = R2 × (0,∞) and φ ∈ C2(Q) such that
w(x, t)− φ(x, t) < w(x0, t0)− φ(x0, t0) for all (x, t) ∈ Q.
Then by definition of w, there exists (xε, tε) ∈ Q such that (xε, tε) → (x0, t0), wε(xε, tε) →
w(x0, t0), and
(wε − φ)(xε, tε) ≥ (wε − φ)(x, t)− ε3
for any (x, t) in a neighborhood of (x0, t0) with size independent of ε. By (7.2.7), we then
have
φ(xε, tε) ≤ maxv,v′∈S1
minb,b′=±1
φ
(xε +
√2ε
2(bv + b′v′), tε − ε2
)+ ε3.
Applying Taylor expansion, we are led to
ε2φt ≤ maxv,v′∈S1
minb,b′=±1
√2ε
2〈∇φ, bv + b′v′〉
+ε2
4〈∇2φ(bv + b′v′), (bv + b′v′)〉
+ o(ε2) at (xε, tε).
(7.2.10)
We denote by I the right hand side evaluated at (xε, tε).
One may first compare 〈∇φ(xε, tε), v〉 and 〈∇φ(xε, tε), v′〉 to determine b or b′. For
example, if
|〈∇φ(xε, tε), v〉| ≤ |〈∇φ(xε, tε), v′〉|, (7.2.11)
111
then we may choose b′0 such that
−|〈∇φ(xε, tε), v′〉| = 〈∇φ(xε, tε), b
′0v′〉
and thus 〈∇φ, bv + b′0v′〉 ≤ 0 for either b = ±1 by (7.2.11). Therefore we have at (xε, tε)
I ≤ maxv,v′
minb
−√
2ε
2|〈∇φ, bv + b′0v
′〉|+ ε2
4〈∇2φ(bv + b′0v
′), (bv + b′0v′)〉
+ o(ε2). (7.2.12)
In what follows, we always assume (7.2.11) and keep the choice b′ = b′0.
Case A. Assume ∇φ(x0, t0) 6= 0, which in turn implies that ∇φ(xε, tε) 6= 0 for all ε > 0
small. We apply Lemma 7.2.5 and Remark 7.2.6 with
p = |bv + b′0v′| ∇φ(xε, tε)
|∇φ(xε, tε)|, ξ = bv + b′0v
′, X = ∇2φ(xε, tε),
and get ∣∣∣∣〈∇2φ(bv + b′0v′), (bv + b′0v
′)〉 − |bv + b′0v′|2⟨∇2φ∇⊥φ|∇φ|
,∇⊥φ|∇φ|
⟩∣∣∣∣≤ M |bv + b′0v
′||∇φ|
|〈∇φ, bv + b′0v′〉|
for any v, v′ ∈ S1. Hence, we have from (7.2.12)
I ≤ ε2∆G1 φ+ εΦ + o(ε2) at (xε, tε), (7.2.13)
where we denote
∆G1 φ =
⟨∇2φ∇⊥φ|∇φ|
,∇⊥φ|∇φ|
⟩= |∇φ| div
(∇φ|∇φ|
),
and
Φ(x, t) = maxv,v′
minb
−√
2
2|〈∇φ(x, t), bv + b′0v
′〉|+ ε
(|bv + b′0v
′|2
4− 1
)∆G
1 φ(x, t)
+εM
4|∇φ(x, t)||bv + b′0v
′||〈∇φ(x, t), bv + b′0v′〉|.
(7.2.14)
Here we denote for any function φ(x, t) ∈ C1(R2) with x = (x1, x2),
∇⊥φ = (−∂φ/∂x2, ∂φ/∂x1) .
112
We next estimate Φ(xε, tε) for ε > 0 sufficiently small. Indeed, we discuss two different cases.
If there exists a subsequence εk such that the maximum attains at v and v′ (depending on
εk) and as k →∞,
either |〈∇φ(xεk , tεk), v〉| → c or |〈∇φ(xεk , tεk), v′〉| → c
for some c > 0, then it is clear that there exists b = ±1 also depending on εk satisfying
|〈∇φ(xεk , tεk), bv + b′0v′〉| > c,
since
maxb|〈∇φ(x, t), bv + b′0v
′〉| ≥ 1
2(|〈∇φ(x, t), v + b′0v
′〉|+ |〈∇φ(x, t),−v + b′0v′〉|)
= max|〈∇φ(x, t), v〉|, |〈∇φ(x, t), b′0v′〉|.
In view of (7.2.14), this yields immediately that
lim supεk→0
Φ(xεk , tεk) ≤ −√
2c
2.
The remaining case is that, for any ε > 0 small, the maximum of Φ is attained at v and v′
such that
|〈∇φ(xε, tε), v〉| → 0 and |〈∇φ(xε, tε), v′〉| → 0 as ε→ 0.
We denote ηε = ∇φ(xε, tε)/|∇φ(xε, tε)| and η⊥ε = ∇⊥φ(xε, tε)/|∇φ(xε, tε)|. Let us take a
modulus of continuity ω such that
max|〈ηε, v〉|, |〈ηε, v′〉| ≤ ω(ε).
Without loss of generality, we may assume
∣∣b′0v′ − η⊥ε ∣∣ ≤ ω(ε).
(Otherwise, we have |b′0v′ + η⊥ε | ≤ ω(ε) and apply the similar argument below.) Then we
may pick b such that ∣∣bv − η⊥ε ∣∣ ≤ ω(ε).
113
It follows that under this choice,∣∣∣∣14 |bv + b′0v′|2 − 1
∣∣∣∣ =1
4|bv − b′0v′|2 ≤
1
2
(|bv − η⊥ε |2 + |b′0v′ − η⊥ε |2
)≤ ω2(ε).
It follows from (7.2.14) that
Φ(xε, tε) ≤ Cεω2(ε) + Cεω(ε)
for some C > 0 independent of ε. Hence, in either case, by (7.2.13), we have
ε2φt ≤ ε2∆G1 φ+ o(ε2) at (xε, tε). (7.2.15)
Dividing the inequality by ε2 and sending ε→ 0, we obtain (7.2.5).
Case B. It remains to show the viscosity inequality
φt(x0, t0) ≤ 0 (7.2.16)
under the conditions that ∇φ(x0, t0) = 0 and ∇2φ(x0, t0) = O.
Case B-1. Suppose that there exists a subsequence ∇φ(xεk , tεk) 6= 0 for k arbitrarily large.
Then we may repeat the argument as in Case A and reach (7.2.15) again. We then obtain
(7.2.16) immediately, since in the present case ∇2φ(xε, tε)→ ∇2φ(x0, t0) = O as ε→ 0.
Case B-2. If ∇φ(xε, tε) = 0 for all ε > 0, then it follows immediately from (7.2.10) that
ε2φt(xε, tε) ≤ Cε2‖∇2φ(xε, tε)‖+ o(ε2),
for some C > 0 independent of ε, which yields (7.2.16).
2. We finally show that w(x, 0) ≤ u0(x). To this end, we construct a game barrier at
every fixed x0 ∈ R2. It is easily seen that for any δ > 0, there exits Kδ > 0 such that
u0(x) ≤ w0(x) = u0(x0) + δ +Kδ|x− x0|2 for all x ∈ R2.
We now play the games with objective function w0 starting from any x in a neighborhood
of x0. In the first step, we have∣∣∣∣∣x− x0 +
√2ε
2(bv + b′v′)
∣∣∣∣∣2
= |x− x0|2 +√
2ε〈x− x0, bv + b′v′〉+ε2
2|bv + b′v′|2.
114
Note that for any v, v′, Player II can pick b′ (or b) so that
〈x− x0, bv + b′v′〉 ≤ 0
for any b (or any b′). This can be done by fixing b′ such that 〈b′v′, x− x0〉 = −|〈v′, x− x0〉|
if |〈v′, x− x0〉| ≥ |〈v, x− x0〉|. Such a strategy yields
∣∣∣∣∣x− x0 +
√2ε
2(bv + b′v′)
∣∣∣∣∣2
≤ 2ε2 + |x− x0|2
for any v, v′. Repeating this strategy of Player II regardless of choices by Player I, we get
wε(x, t) ≤ w0(x) + 2Kδt.
This means w(x0, 0) ≤ w0(x0) ≤ u0(x0) + δ. We conclude the proof by letting δ → 0.
The following comparison result is an immediate consequence of Theorem 7.2.4 and the
proof is due to [25, 34].
Corollary 7.2.7 (Comparison theorem). Suppose that u is the viscosity solution of (MCF)
and w is the upper relaxed limit of the game value wε associated to the modified game. Then
w ≤ u in R2 × (0,∞).
115
7.3 GAME-THEORETIC PROOFS OF CONVEXITY-PRESERVING
PROPERTIES
7.3.1 Convexity-preserving for the solutions to the normalized parabolic p-
Laplace equations
Recall the DPP related to the normalized parabolic p-Laplace equations is:
uε(x, t) =α
2supv∈Bε
uε(x+ v, t− σpε2) +α
2infv∈Bε
uε(x+ v, t− σpε2)
+ β
v∈Bε
uε(x+ v, t− σpε2) dv + ε2f(x),(7.3.1)
uε(x, t) = u0(x) for all x ∈ Rn and t < ε2. (7.3.2)
We next use the game to prove the following.
Theorem 7.3.1. Assume that u0 and f are both convex and Lipschitz continuous in Rn.
Suppose that the function uε satisfying (7.3.1)–(7.3.2) converges to the solution u of (PL).
Then the solution u(x, t) is convex in x for any t ≥ 0; namely,
1
2u(x− h, t) +
1
2u(x+ h, t) ≥ u(x, t) (7.3.3)
for any x, h ∈ Rn and t ≥ 0.
Proof. It suffices to show (7.3.3) with u replaced by uε. Let us take a look at the first game
step:
uε(x± h, σpε2) =α
2supv∈Bε
u0(x± h+ v) +α
2infv∈Bε
u0(x± h+ v)
+ β
v∈Bε
u0(x± h+ v) dv + ε2f(x± h),(7.3.4)
116
We assume that the infimum appearing in each equalities in (7.3.4) above is attained respec-
tively at v1, v2 ∈ Bε and also assume that supv∈Bεu0(x + v) = u0(x + v) for some v ∈ Bε.
(These can be rigorously justified by taking errors into account.) Hence, we have
uε(x+ h, σpε2) ≥ α
2u0(x+ h+ v) +
α
2u0(x+ h+ v1)
+ β
v∈Bε
u0(x+ h+ v) dv + ε2f(x+ h);
uε(x− h, σpε2) ≥ α
2u0(x− h+ v) +
α
2u0(x− h+ v2)
+ β
v∈Bε
u0(x− h+ v) dv + ε2f(x− h).
By convexity of u0 and f , we get
uε(x+ h, σpε2) + uε(x− h, σpε2)
≥ αu0(x+ v) + αu0
(x+
v1 + v2
2
)+ 2β
v∈Bε
u0(x+ v) dv + 2ε2f(x)
≥ α supv∈Bε
u0(x+ v) + α infv∈Bε
u0(x+ v) + 2β
v∈Bε
u0(x+ v) dv + 2ε2f(x).
(7.3.5)
Since, by definition, the right hand side is just 2uε(x, σpε2), we have
uε(x+ h, σpε2) + uε(x− h, σpε2) ≥ 2uε(x, σpε
2).
We repeat the argument and end up with
uε(x+ h, t) + uε(x− h, t) ≥ 2uε(x, t)
for any t ≥ 0. Sending ε→ 0, we get the convexity desired.
117
7.3.2 Convexity-preserving for level sets of the mean curvature flow equations
For future use, let D0c and E0
c respectively denote the open and closed c-superlevel set of u0
for any c ≥ a; that is,
D0c = x ∈ R2 : u0(x) > c; E0
c = x ∈ R2 : u0(x) ≥ c.
Then we have D0c ⊂ D0
d and E0c ⊂ E0
d for any c ≥ d. Moreover, we have D0c =
⋃d>cE
0d .
Lemma 7.3.2 (Monotonicity). Suppose that u0 satisfies (A). Assume that the superlevel set
E0c for each c ≥ a is convex. Let uε be the value function associated to the game. Then
uε(x, t) ≤ uε(x, s) for all x ∈ R2, t ≥ s ≥ 0 and ε > 0. (7.3.6)
In particular, u(x, t) ≤ u(x, s) for all x ∈ R2, t ≥ s ≥ 0.
Proof. Let us fix ε > 0. We begin our proof by claiming that for every x ∈ R2 and any
v ∈ S1, there exists b = ±1 such that
u0
(x+√
2εbv)≤ u0(x). (7.3.7)
Indeed, assume that for some v ∈ S1 and some λ > u0(x),
u0
(x−√
2εv), u0
(x+√
2εv)≥ λ.
By convexity of the superlevel set E0λ, x = 1
2(x−√
2εv)+ 12(x+√
2εv) ∈ E0λ which contradicts
the assumption that u0(x) < λ.
Now let us incorporate the claim in our games. Notice that for any game position z(s;x)
from an arbitrary x ∈ R2 after time s > 0, depending on the choices v1, b1, v2, b2, . . . , vN , bN
of both players (N = [s/ε2]), Player II may use the strategy described above to ensure
maxv∈S1
minb=±1
u0
(z(s;x) +
√2εbv
)≤ u0(z(s;x)) for any v ∈ S1.
By applying this strategy from step N to N ′ = [t/ε2], we obtain
maxvN+1
minbN+1
. . .maxvN′
minbN′
u0
(z(s;x) +
√2ε
N ′∑i=N+1
bivi
)≤ u0(z(s;x)).
118
Taking the extrema on both sides of the inequality above over bN , vN , bN−1, vN−1, . . . , b1
and v1 in order, we get (7.3.6) by definition. It follows immediately that u(x, t) ≤ u(x, s) for
all t ≥ s, by passing to the limit as ε→ 0 in (7.3.6) with application of Theorem 7.2.1.
Theorem 7.3.3 (Convexity of level sets). Suppose that u0 satisfies (A). Assume that each
superlevel set E0c (c ≥ a) of u0 is convex. Let uε be the value function associated to the game.
Then every superlevel set of uε(·, t) is almost convex for any t ≥ 0 and ε > 0 in the sense
that there exists a modulus ω of continuity depending on u0 such that for any x, y ∈ R2 and
t ≥ 0,
uε(x+ y
2, t
)≥ c− ω(ε) (7.3.8)
provided that uε(x, t) ≥ c and uε(y, t) ≥ c. In particular, all superlevel sets of the solution
u(·, t) of (MCF) are convex for every t ≥ 0.
Proof of Theorem 7.3.3. We assume x 6= y, since otherwise the statements are trivial. Since
uε(x, t) ≥ c and uε(y, t) ≥ c, we have uε(x, s) ≥ c and uε(y, s) ≥ c for s ≤ t as well in virtue
of the monotonicity shown in Lemma 7.3.2. (In particular, u0(x) ≥ c and u0(y) ≥ c.)
Then for any s ≤ t, there must exist maximizing strategies SIx,s and SIy,s of Player I such
that regardless of the choices of Player II, we have u0(z(s;x)) ≥ c and u0(z(s; y)) ≥ c if SIx,s
and SIy,s are applied respectively in the games starting from x and y.
We next consider the game started from (x+y)/2. In this case, Player I has the following
possible of move: he keeps choosing v = (x−y)/|x−y| until the game position enters B√2ε(x)
or B√2ε(y). Here Br(ξ) denotes the open ball centered at ξ ∈ Rn with radius r. Without
loss of generality, suppose that Player II chooses to let z(τ ; (x+ y)/2) ∈ B√2ε(x) after time
τ(≤ t). Then Player I may use SIx,s with s = t − τ to bring the game position to ξ ∈ Rn,
which depends on the response of Player II to SIx,s. However, the same strategy of Player II
may send x to z(s;x), which is in the√
2ε-neighborhood of ξ. Since u0(z(s;x)) ≥ c, we get
the following estimate:
u0(ξ) ≥ u0(z(s;x))− ω0(√
2ε) ≥ c− ω0(√
2ε).
119
The remaining case is that Player II may choose to let the game position wander away
from the neighborhoods of x and y. But in this case the final position η must still stay on
the line segment between x and y and therefore
u0(η) ≥ c,
due to the facts that u0(x) ≥ c and u0(y) ≥ c and the assumption that superlevel sets of u0
are convex.
Since each of game outcomes is just for one possible strategy of Player I and his optimal
strategy should be even better, we end up with (7.3.8) with ω(x) set to be ω0(√
2x).
Now in order to prove the statement concerning u, we only need to take the limit using
Theorem 7.2.1. More precisely, for any δ > 0, there exists ε > 0 such that
uε(x, t) ≥ c− δ and uε(y, t) ≥ c− δ.
Our argument above yields
uε(x+ y
2, t
)≥ c− δ − ω(ε).
We conclude the proof by letting δ → 0.
Remark 7.3.4. In the theorem above, we do not assume the concavity of u0 itself but the
convexity of its level sets. Our convexity result is therefore only for the convexity of level
sets as well. Also, to study convexity of a particular level, it is not restrictive to assume that
all level sets of u0 are convex curves. Note that changing the other level sets of u0 will not
affect the evolution of the particular level set in question [34, 25, 38].
Remark 7.3.5. An alternative method, without assuming the convexity of all level sets but
strict convexity of the initial level set, is to investigate the convexity of the level sets of
the solution to (SP) with Ω (strictly) convex. The stationary problem has a similar game-
theoretic approximation [78] but the value function uε(x) in this case is defined to be the
optimized first exit time of the domain starting from x ∈ Ω. It is not difficult to see that
the argument in the proof of Theorem 7.3.3 applies exactly to this case as well. A new PDE
proof for this problem was recently given in [12].
120
7.3.3 Convexity-preserving for the solutions to the mean curvature flow equa-
tions
We next show the convexity preserving property for the solution itself. Due to an opposite
choice of orientation, we here discuss instead the equivalent concavity preserving property,
which, under the assumption (A), means that the solution u(x, t) of (MCF) is concave with
respect to x in its open superlevel set
Dta := x ∈ Rn : u(x, t) > a
if u0 is concave in Ka.
Theorem 7.3.6 (Concavity preserving of the solution). Suppose that u0 satisfies (A). As-
sume that u0 is concave in Ka. Let uε and wε be respectively the value functions of the
original game and the modified game. Then uε(x, t) satisfies
uε(x+ h, t) + uε(x− h, t) ≤ 2wε(x, t), (7.3.9)
for any x, h ∈ R2, t > 0 and ε > 0 if
uε(x+ h, t) > a and uε(x− h, t) > a. (7.3.10)
Moreover, the solution u(x, t) of (MCF) is concave with respect to x in Dta for any t ≥ 0.
To show this result, we need the following elementary result related to our games.
Lemma 7.3.7. Let Ω be an open set in R2. Suppose that w,W ∈ C(R2) satisfy
(1) w > a in Ω for some a ∈ R and w = a on R2 \ Ω;
(2) for all x, h ∈ R2 such that x± h ∈ Ω,
w(x+ h) + w(x− h) ≤ W (x). (7.3.11)
121
Then for any constant λ > 0,
maxv∈S1
minb=±1
w(x+ h+ λbv) + maxv∈S1
minb=±1
w(x− h+ λbv)
≤ maxv1,v2∈S1
minb1,b2=±1
W
(x+
λ
2(b1v1 + b2v2)
) (7.3.12)
provided that
maxv∈S1
minb=±1
w(x+ h+ λbv) > a and maxv∈S1
minb=±1
w(x− h+ λbv) > a. (7.3.13)
Proof. We take v+ and v− such that
maxv
minbw(x± h+ λbv) = min
bw(x± h+ λbv±)(> a), (7.3.14)
Let us next pick b± such that
W
(x+
λ
2(b+v+ + b−v−)
)= min
b1minb2
W
(x+
λ
2(b1v+ + b2v+)
),
which implies that
W
(x+
λ
2(b+v+ + b−v−)
)≤ max
v1,v2minb1,b2
W
(x+
λ
2(b1v1 + b2v2)
). (7.3.15)
On the other hand, by (7.3.14), we have
maxv∈S1
minb=±1
w(x± h+ λbv) ≤ w(x± h+ λb±v±). (7.3.16)
Due to (7.3.16) and (7.3.13) we have x ± h + λb±v± ∈ Ω, since w(x ± h + λb±v±) > a. We
finally obtain (7.3.12) by combining (7.3.11), (7.3.15) and (7.3.16).
122
Proof of Theorem 7.3.6. Fix ε > 0 and set Dta := x ∈ Rn : uε(x, t) > a. We first apply
Lemma 7.3.7 with λ =√
2ε, Ω = Ka and
w(x) = u0(x) and W (x) = 2u0(x).
Since it is clear that (7.3.11) holds due to the concavity of u0 in Ka, we get
maxv∈S1
minb=±1
w(x+ h+√
2εbv) + maxv∈S1
minb=±1
w(x− h+√
2εbv)
≤ maxv1,v2∈S1
minb1,b2=±1
W
(x+
√2ε
2(b1v1 + b2v2)
)
if both terms on the left hand side are greater than a. This amounts to saying that
uε(x+ h, ε2) + uε(x− h, ε2) ≤ 2wε(x, ε2) (7.3.17)
for all x, h provided that uε(x±h, ε2) > a. Noticing that uε(x, ε2) and wε(x, ε2) are Lipschitz
continuous in x, due to Proposition 7.2.3 and [78, Appendix B], we can continue using Lemma
7.3.7 with λ =√
2ε, Ω = Dε2
a and
w(x) = uε(x, ε2) and W (x) = 2wε(x, ε2).
Analogously, we obtain
uε(x+ h, 2ε2) + uε(x− h, 2ε2) ≤ 2wε(x, 2ε2)
if uε(x± h, 2ε2) > a, thanks to the dynamic programming principle:
uε(x+ h, 2ε2) = maxv∈S1
minb=±1
uε(x+ h+√
2εbv, ε2);
uε(x− h, 2ε2) = maxv∈S1
minb=±1
uε(x− h+√
2εbv, ε2);
wε(x, 2ε2) = maxv1,v2∈S1
minb1,b2=±1
wε
(x+
√2ε
2(b1v1 + b2v2), ε2
).
We keep iterating the arguments above and eventually get
uε(x+ h,Nε2) + uε(x− h,Nε2) ≤ 2wε(x,Nε2), (7.3.18)
123
for any x, h ∈ Rn satisfying (7.3.10). This is exactly the desired inequality (7.3.9). Since
ε > 0 is arbitrary in (7.3.18), by passing to the limits as ε→ 0 and applying Theorem 7.2.1,
we get
u(x+ h, t) + u(x− h, t) ≤ 2w(x, t)
provided that x ± h ∈ Dta. The concavity preserving property for the solution u follows
immediately from the comparison that w ≤ u in Corollary 7.2.7.
The concavity or convexity preserving property does not precisely hold on the discrete
level in general. In other words, one cannot expect in general that
uε(x− h, t) + uε(x+ h, t) ≤ 2uε(x, t)
for all x, h ∈ R2, t > 0 and ε > 0. We give an example to show this.
Example 7.3.8. Let u0 be Lipschitz and concave in R2. Suppose that the level set x ∈ R2 :
u0(x) = 0 consists of the positive axes. We assume that u0 > 0 in the first quadrant. Let
x0 =(√
2ε/2,√
2ε/2)
and h = (√
2ε/2,−√
2ε/2).
When ε > 0 is taken small, it is easily seen that uε(x0± h, ε2) = 0, since the maximizing
choices of v for Player I to start the game at x0 +h or x0−h can be the ones along the axes.
On the other hand, at the point x0, the outcome after one step is always negative no matter
what choice Player I makes, i.e., uε(x0, ε2) < 0. We therefore have
uε(x0 + h, ε2) + uε(x0 − h, ε2) ≥ 2uε(x0, ε2),
although u0 is concave. One may easily modify this example to have u0 also satisfy (A).
124
7.3.4 Convexity-preserving with Neumann boundary condition of the mean cur-
vature flow equations
Our game-theoretic approach to convexity can be extended to the Neumann problems as
well. In this section, we assume that Ω is a smooth bounded convex domain in R2 and ν(x)
denote the unit outward normal to ∂Ω. We consider the Neumann boundary problem for
the level set curvature flow equation in Ω:
(NP)
ut − |∇u| div
(∇u|∇u|
)= 0 in Ω× (0,∞), (7.3.19)
∇u(x, t) · ν(x) = 0 for x ∈ ∂Ω and t > 0, (7.3.20)
u(x, 0) = u0(x) for all x ∈ Ω. (7.3.21)
Hereafter we assume an analogue of (A):
(A1) u0 is a bounded Lipschitz continuous function in Ω satisfying that u0 ≥ a and u0− a has
compact support Ka ⊂ Ω for some constant a ∈ R.
We refer [43, 109] for existence and uniqueness for viscosity solutions of this problem.
7.3.4.1 Billiard games for the Neumann condition We first recall a billiard game
interpretation for this problem in [42]. For any t ≥ 0, x ∈ Ω and v ∈ S1, let St(x, v) ∈ Ω
denote the position of billiard motion starting from x with initial direction v and with
distance of travel equal to t; see [42, Definition 2.1].
In view of [42, Lemma 2.3], we have
St(x, v) = x+ tv − αt(x, v), (7.3.22)
where αt(x, v) ∈ R2 is called a boundary adjustor, satisfying |αt(x, v)| ≤ 2t. More precisely,
we may write
αt(x, v) =∞∑k=0
ckν(yk), (7.3.23)
with ck ∈ R, yk ∈ ∂Ω ∩Bt(x).
We apply this billiard dynamics to our game setting by slightly changing the game rules
introduced in Section 7.2.2. We restrict the starting point x to be in Ω and substitute the
rule (3) with the following:
125
(3)’ The marker is moved from the present state x to S√
2ε(x, bv).
We may define the value function uε to be in the same form of (7.2.3) by establishing a new
state equation: zk+1 = S√
2ε(zk, bkvk), k = 0, 1, . . . , N − 1;
z0 = x,(7.3.24)
where vk ∈ S1 and bk = ±1.
Theorem 7.3.9. [42, Theorem 1.1] Assume that Ω is a smooth bounded convex domain in
R2. Let uε be the value function associated to the game above. Then uε converges, as ε→ 0,
to the unique viscosity solution of (NP) uniformly on compact subsets of Ω× [0,∞).
Remark 7.3.10. There is more than one way to generate the boundary condition of Neumann
type in games. One may simply adopt direct constraints of game trajectories in the domain
to establish tug-of-war game interpretations of Neumann boundary problem for the infinity
Laplacian in [23, 6]. Their method relies on the special structure of the infinity Laplacian and
is not applicable in our present case. A different but more relevant approach is presented in
[29] for the Neumann problems for a general class of elliptic and parabolic equations without
assuming the boundedness and convexity of the domain, where the author uses boundary
projection instead of our billiard motion. The game convergence result in [29] does not
include (NP), but it is possible to adapt the argument to our case.
7.3.4.2 Convexity of level sets As in the preceding sections, we take for any t ≥ 0
Dtc = x ∈ Ω : u(x, t) > c, Et
c = x ∈ Ω : u(x, t) ≥ c and Γtc = Etc \Dt
c.
We aim to show that for any fixed c ≥ a, E0c being convex implies that Et
c is convex for any
t ≥ 0. We assume
(A2) E0c is convex for any c ≥ a.
It is clear that under the assumption (A2), for any x ∈ Ω with u0(x) = c ≥ a, there exists
a supporting line Lx of E0c passing through x; that is, Lx divides R2 into two half planes,
only one of which has nonempty intersection with D0c . We denote by ξ(x) the outward unit
normal to the half plane containing D0c .
126
We will also use a compatibility condition as given below:
Definition 7.3.11 (Compatibility condition). For any u0 satisfying (A2), we say u0 is
(weakly) compatible with the Neumann boundary condition (7.3.20) if for any τ > 0 small
and x0 ∈ ∂Ω there exists a supporting line Lx0 with normal ξ(x0) satisfying
u0(Sτ (x0, v)) ≤ u0(x0) (7.3.25)
for all v ∈ S1 with 〈v, ξ(x0)〉 ≥ 0.
Theorem 7.3.12 (Convexity preserving for the Neumann problem). Assume that Ω is a
smooth bounded convex domain in R2. Let u be the unique viscosity solution of (NP) with
u0 satisfying (A1) and (A2). Assume that u0 also satisfies the compatibility condition as in
Definition 7.3.11. Then the superlevel sets Etc of u(·, t) for any t ≥ 0 are convex for any
t ≥ 0.
In order to prove this theorem, we again need a monotonicity result similar to Lemma
7.3.2.
Lemma 7.3.13 (Monotonicity for the Neumann problem). Suppose that u0 satisfies (A1)
and (A2). Assume that u0 also satisfies the compatibility condition. Let uε be the value
function associated to the billiard game described above. Then
uε(x, t) ≤ uε(x, s) for all x ∈ Rn, t ≥ s ≥ 0 and ε > 0. (7.3.26)
In particular, the solution u is monotone in time, i.e., u(x, t) ≤ u(x, s) for all x ∈ Rn,
t ≥ s ≥ 0.
Proof. Let us fix ε > 0. We claim this time that for every x ∈ Rn,
supv∈S1
infb=±1
u0
(S√
2ε(x, bv))≤ u0(x). (7.3.27)
Indeed, for any v ∈ S1, one may pick b = ±1 such that 〈ξ(x), bv〉 ≥ 0. If the billiard
motion does not touch ∂Ω in the initial direction bv, then by convexity of the superlevel set
with level u0(x), we get
u0
(S√
2ε(x, bv))≤ u0(x).
127
In the case that the billiard trajectory hits ∂Ω, we have
S√
2ε(x, bv) = S√
2ε−τ (y, bv), (7.3.28)
where y = x + τ bv ∈ ∂Ω denotes the first hitting position with τ ≥ 0. For the same reason
as above, y must belong to the lower level of u0, i.e., u0(y) ≤ u0(x). We next apply the
compatibility condition to get
u0(S√
2ε−τ (y, bv)) ≤ u0(y),
which by (7.3.28) yields
u0(S√
2ε(x, bv)) ≤ u0(x).
The rest of the proof, similar to that in the proof of Lemma 7.3.2 consists in the iteration
of (7.3.27) and passing to the limit as ε→ 0 with application of Theorem 7.3.9. We omit it
and refer the reader to the proof of Lemma 7.3.2.
The proof of Theorem 7.3.12 is essentially the same as that of Theorem 7.3.3. However, in
the proof of Theorem 7.3.3 we used the Lipschitz continuous dependence of game strategies
on the initial positions and directions, which is not clear in the current case for the billiard
motion. We thus provide a slightly different proof by choosing a subsequence of game values.
This proof also works for Theorem 7.3.3.
Proof of Theorem 7.3.12. We take x, y ∈ Ω with x 6= y and u(x, t), u(y, t) ≥ c for some t ≥ 0
and c ≥ a. We aim to show that u(x+y2, t) ≥ c. We set εm = |x−y|
2√
2mfor any positive integer
m. In view of Theorem 7.3.9, for any δ > 0 small, there exists m sufficiently large, such that
uεm(x, t), uεm(y, t) ≥ c− δ.
We then have uε(x, s) ≥ c− δ and uε(y, s) ≥ c− δ for s ≤ t as well due to Lemma 7.3.13.
Then for any s ≤ t, there must exist maximizing strategies SIx,s and SIy,s of Player I such
that regardless of the choices of Player II, we have u0(z(s;x)) ≥ c−2δ and u0(z(s; y)) ≥ c−2δ
if SIx,s and SIy,s are applied respectively in the games starting from x and y.
We next consider the game started from (x + y)/2. If Player I keeps choosing v =
(x − y)/|x − y| until the game position reaches x or y. Without loss of generality, suppose
128
that Player II chooses to let z(τ ; (x + y)/2) = x after time τ(≤ t). Then Player I may use
SIx,s with s = t− τ to bring the game position to ξ ∈ Rn, which depends on the response of
Player II to SIx,s. This yields
u0(z(t; (x+ y)/2)) = u0(z(s;x)) ≥ c− 2δ.
Player II may choose to let the game position wander away from the neighborhoods of x and
y. In this case the final position η must still stay on the line segment between x and y and
therefore
u0(η) ≥ c− 2δ,
due to the assumption that superlevel sets of u0 are convex.
Since the above game estimate is for a fixed strategy of Player I, we get
uεm(x+ y
2, t
)≥ c− 2δ.
Thanks to Theorem 7.3.9, we conclude the proof by passing to the limit as m→∞ and then
δ → 0.
Remark 7.3.14. As mentioned in Remark 7.3.10, it is possible to have a game approximation
result without the boundedness and convexity assumptions on Ω, following [29]. Therefore
one may also expect that the convexity preserving property still holds without assuming
the boundedness and convexity of Ω. In fact, without convexity of Ω, we can use the same
argument to prove that Etc preserves convexity relative to Ω under the assumption (A1),
(A2) and the compatibility condition. To be more precise, we have
u
(x+ y
2, t
)≥ c
whenever x, y ∈ Ω and t ≥ 0 satisfy u(x, t), u(y, t) ≥ c and kx+(1−k)y ∈ Ω for all k ∈ [0, 1].
129
We finally make some remarks on the compatibility condition in Definition 7.3.11. Let
us discuss a smooth special case. Suppose u0 is of class C2 and concave in Ω. Then u0 is
compatible with the Neumann boundary condition if there is σ > 0 such that
〈∇u0(x0), ν(y)〉 ≥ 0 and ∇2u0(x0) ≤ −σI. (7.3.29)
for any x0 ∈ ∂Ω and any y ∈ Bσ(x0) ∩ ∂Ω.
Indeed, in this case one may choose ξ(x0) = −∇u0(x0)/|∇u0(x0)|. To simplify notation,
we write ξ0 instead of ξ(x0). For any v ∈ S1 with 〈v, ξ0〉 ≥ 0, we write
v = τ1ξ0 + τ2ξ⊥0 ,
where ξ⊥0 is the unit vector orthogonal to ξ0 and τ1 ≥ 0, τ2 ∈ R satisfy τ 21 + τ 2
2 = τ 2. We
apply Taylor expansion to get
u0(Sτ (x0, v)) = u0(x0) + τ1〈∇u0(x0), ξ0〉 − 〈∇u0(x0), ατ (x0, v)〉
+1
2〈∇2u0(x0)(τv − ατ (x0, v)), (τv − ατ (x0, v))〉+ o(|τv − ατ (x0, v)|2),
which, due to (7.3.29) and (7.3.23), implies (7.3.25) for τ sufficiently small.
It is worth pointing out that the curvature flow Γt fails to preserve convexity in general
if the compatibility condition as in Definition 7.3.11 is not satisfied.
Example 7.3.15. Consider the special case when Ω = (−1, 1)×R and Γt can be expressed as
the graph of a function y = v(x, t) for (x, t) ∈ [−1, 1]× [0,∞), we deducevt −
vxx1 + v2
x
= 0 in (−1, 1)× (0,∞), (7.3.30)
vx(−1, t) = vx(1, t) = 0 for all t > 0, (7.3.31)
v(x, 0) = v0(x) for all x ∈ [−1, 1]. (7.3.32)
It is known [87, 9, 10] that there still exists a unique viscosity solution of this problem for
any Lipschitz continuous function even when v0 does not fulfill the compatibility condition.
Suppose that v0 is an even Lipschitz function on [−1, 1] with
(v0)x(1) = −(v0)x(−1) > 0.
Then for any t ≥ 0, the unique solution v(x, t) must also be even in x.
130
On the other hand, one may extend v0 to a Lipschitz function on R in a periodic manner
and then solve the corresponding Cauchy problem in R× [0,∞). It is clear that the solution
is still space periodic for any time. Moreover, it is shown by Ecker and Huisken [32] that
the solution v(x, t) is smooth for any t > 0, which implies that vx(±1, t) = 0 and v cannot
be convex around x = ±1. This means that the restriction of v on [−1, 1] × [0,∞) is the
unique solution of (7.3.30)–(7.3.32). The failure of being convex near x = ±1 remains with
such a restriction.
For the same reason as explained above, one cannot in general expect the viscosity
solution of (NP) itself to be convexity preserving without assuming a more restrictive com-
patibility condition.
131
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